This is a very long answer, but it does suggest several ways to make ruling a galactic government with slower than light travel less impossible, and perhaps even possible.
1) increase the life span of humans and/or aliens living in the galactic government.
If it takes thousands to tens of thousands to hundreds of thousands to millions to tens of millions of years for sub light space ships to travel across a galaxy, a human ruled galaxy would not be possible until at least that far in the future of when those humans were as advanced as Earth in 2018. Humans wouldn't have even reached the farthest parts of their galaxy until that long after having invented interstellar ship travelling at some fraction of the speed of light, which won't happen until they are more advanced than Earth in the year 2018.
So I would certain hope that by then those humans would have made the discoveries and inventions that seem the most desirable of all to me - methods of extreme life extension or even immortality.
You may remember that famous science fiction series Cities in Flight by James Blish. In the chronologically first novel in the series, Earth people discover both the "spindizzy" that makes faster than light travel possible and the first of the antiagathic (anti death) drugs that enable humans to live for centuries or millennia. In the novel titled They Shall Have Stars (1956) or Year 2018!.
Although it doesn't look like anti death drugs will be discovered this year, we can hope that they will be discovered long before 2118, and long before 3018, and long before 20,018, and long before 200,018, and long before 2,000,018, which would be almost the earliest possible date for a galactic government to be established.
So if people expect to live for centuries, millennia, or even much longer, it will make it a lot easier to have a galactic government where messages travel at the speed of light and space warships travel much slower than that.
2) Make your interstellar government hierarchical, and/or federal, and/or feudal, in structure.
It seems to me that an interstellar government would probably be hierarchical long before it ruled an entire galaxy. Even the government of a single star system should be highly hierarchical if there are many artificial space habitats in the system.
Possibly the ruler of a single habitat would be a habicrat ("habitat ruler") and the ruler of ten habitats would be a "habicrat of habicrats", or "second level habicrat", or "habicrat to the second power", and the ruler of 100 habitats would be a "habicrat of habicrats of habicrats", or "third level habicrat", or "habicrat to the third power", and so on up to the stellacrat or "star ruler" ruling the entire star system.
If a stellacrat rules an entire star system, a single, binary, or multiple star system and its planets and space habitats, a "stellacrat of stellacrats" or "second level stellacrat" would rule ten star systems, a "third level stellacrat" would rule 100 star systems, and so on and on up to a "seventh level stellacrat" ruling a million star systems. Since galaxies have about one million stars to one trillion stars, a "galacticrat" or "galaxy ruler" would be the equivalent of a "seventh level stallacrat" to a "thirteenth level stellacrat".
Since the popular imagination tends to confuse outer space and heaven, it is possible that the immediate subordinates of a stellacrat and/or those of a galacticrat might be called after various levels in the medieval hierarchy o f angels.
In a star system with many space habitats, the immediate subordinates of a stellacrat might be 10 "stellar Seraphim", with 100 "Stellar Cherubim" below them, 1,000 "Stellar Thrones", 10,000 "Stellar Dominions", 100,000 "Stellar Virtues", 1,000,000 "Stellar Powers", "10,000,000 "Stellar Principalities", 100,000,000 "Stellar Archangels", and "1,000,000,000 "Stellar Angels", and below them various levels of habicrats.
In a galaxy with billions of star systems, the immediate subordinates of a galacticrat might be 10 "Galactic Seraphim", with 100 "Galactic Cherubim" below them, 1,000 "Galactic Thrones", 10,000 "Galactic Dominions", 100,000 "Galactic Virtues", 1,000,000 "Galactic Powers", "10,000,000 "Galactic Principalities", 100,000,000 "Galactic Archangels", and 1,000,000,000 "Galactic Angels", and below them various levels of stellacrats.
3) Have local elections within a star system, and also elections on an interstellar scale if space ship speeds and the life spans of the citizens allow it.
Within a star system, it would only take hours for news to travel at the speed of light, unless it was a binary or multiple star system with widespread stars or the outer cometary halo is colonized, in which case it could take months for news to travel at the speed of light.
Thus it is possible for elections to happen throughout a star system and for those elected to take office relatively soon after the election, weeks, months or possibly years at sublight speeds.
But having elections throughout an area that is ten light years across means that the news of the political platforms could take ten years to reach the voting places before the election, and ten years for the results to reach every place, and then possibly decades for those elected to office in the central government to reach the location of the central government.
Thus people would likely be voting for a government that would take office in fifty years or a century after the start of the election, and would have to be very long lived for that to seem normal to them.
And if the area of voting was 100 light years across that would mean voting for a government that would take office centuries or millennia in the future, so the voters would have to be much more longer lived for that to be acceptable to them.
In a galaxy like ours with a disc about 100,000 light years in diameter, it would take 50,000 yeaes for news of a political party's platform to reach the outer rim if the convention was held in the center of the Galaxy, and 50,000 more years for news of the election results to reach the center of the galaxy, and who knows how many hundreds of thousands or millions of years for those elected to the central government to reach the galactic capital. So galaxy wide elections would probably be held at least a million years apart, and the electorate would have to be extremely long lived for that to seem like a short enough interval to them.
Thus it is possible that elections would only be held in a star system wide scale. And probably also on many lower levels below the system wide government.
4) Even within a star system, there should be various levels of government.
The United States of America has basically four levels of government (ignoring some exceptions like over 500 tribal governments): federal, state, county, and municipal, and holds elections for all four levels.
According to Wikipedia's List of Administrative Divisions by Country, there is a great variety in the number of levels of administrative divisions between different countries. Usually, countries with larger populations and/or areas have more levels of administrative divisions that countries with smaller populations and/or areas, but there are many exceptions.
The 10 largest countries by area are: Russia (3 levels of administrative divisions), Canada (3), China (5), the USA (3), Brazil (2), Australia (2), India (4), Argentina (2), Kazakhstan (4), and Algeria (2).
The 10 smallest countries by area are Vatican City (zero levels of administrative divisions), Monaco (2), Nauru (1), Tuvalu (1), San Marino (1), Lichtenstein (1), the Marshall Islands (1), Saint Kitts and Nevis (2), the Maldives (2), and Malta (2).
The 10 largest countries by population are: China (5 levels of administrative divisions), India (4), the USA (3), Indonesia (4), Brazil (2), Pakistan (4), Nigeria (2), Bangladesh (4), Russia (3), and Japan (4).
The 10 smallest countries by population are: Vatican City (zero levels of administrative divisions), Nauru (1), Tuvalu (1), Palau (1), San Marino (1), Lichtenstein (1), Monaco (2), Saint Kitts and Nevis (2), The Marshall Islands (1), and Dominica (1).
Note that Monaco with an area of 2.02 square kilometers and a population of 38,300, and Brazil with a land area of 8,460,415 square kilometers and a population of 209,550,000, both have two levels of administrative subdivisions.
Monaco has 1 Commune and 10 quartiers, while Brazil has 26 estados and one distrito Federal and 5,569 municipios.
So if the Earth was united by negotiation and/or conquest at the present time, any of the present national governments that continued to function would become another level of administrative division, and thus in various parts of the world there might be between one and six levels of administrative divisions below the world government. Of course it is possible that intermediate levels of administrative divisions might be formed, such as continental governments. If each continent had a continental government then there would be between two and seven levels of administrative divisions below the world government in various regions.
And if the solar system becomes populated by billions and trillions and quadrillions of people living in many millions of space habitats, there could be ten or more levels of administrative divisions, and there could be local elections in every administrative division.
And if people live really long extended lifetimes it might be possible to hold elections in administrative divisions that cover interstellar space and many star systems. The longer people live, the more they might tolerate infrequent and long lasting elections. They would have to have lifespans of millions of years for galaxy wide elections to be possible.
5) Local elected governments within star systems should have the most power over the lives of their citizens, and thus sometimes anger those citizens the most.
So the plan for a galactic government, since it may not be certain if it is possible to hold elections in larger areas than a single star system, would have to be to have a many leveled government within each star system with elections at each level, and most of the things that matter to people should be decided at one or more of the levels within the star system.
Thus when people get dissatisfied with their government, it will be a local government that they are dissatisfied with. So they will overthrow the local government and not the central government of the galaxy, and they will overthrow the local government by electing a new administration instead of by a revolt which could turn into a revolt against the central government of the galaxy.
6) Keep military force from higher levels of government within each star system to protect against invaders and to crush rebellion.
And presumably there would be units of the central government's space navy stationed in every star system to stop any attempts to revolt against the central government, and to guard the star system against any possible invasion.
These units would be sent from the star system with capital of the governmental administrative division immediately above the star system, and presumably the families of the crews would be kept in suspended animation or something until the crews returned from their tour of duty. The families would also be possible hostages if the garrisons revolted.
The star system with the government center of this next highest administrative division would be garrisoned by space navy units sent from the star system with the government center of the next highest administrative division above that, and so on and so on.
If each administrative division rules X administrative divisions below it, it would have to send X number of fleets to the government centers of the administrative divisions below it. Thus the resources needed by each star system that was the center of an administrative division would be the same.
At least that would be the cases if there was a faster than light drive, one fast enough that a voyage across the galaxy would take less than a year, for example.
If spaceships travel at speeds less than the speed of light, maybe 99.99 percent of the speed of light, or 10 percent of the speed of light, or 1 percent of the speed of light, or maybe even only 0.1 percent of the speed of light, the higher up star systems would have to devote more personnel and resources to sending fleets to their immediate subordinates.
The longer the distance the fleets had to travel, the longer the trips will take. The longer that the people in this galactic civilization live, the longer the missions to garrison other worlds they can tolerate. If people can tolerate a round trip time of X years, and if the transit time is Y years, X years minus 2 Y years equals Z years, the number of years they can spend garrisoning a different solar system.
Suppose that a local government sends a garrison to a star system five light years away, at half the speed of light, and a total trip time of 10 years. If the members of the garrison can tolerate 30 years away from home, Y and Z will both be one third of X, and at anyone time there should be one garrison on duty, one returning, and one on route to the destination star, so if they do that for nine other star systems they will have to support twenty seven garrisons and fleets at any one time.
And if Y equals 10 times Z, X will equal 21 times Z, so the system sending the garrisons and fleets will have to support 21 times as many fleets and garrisons as the number of systems it garrisons.
And if Y equals 100 times Z, X will equal 201 times Z, so the system sending the garrisons and fleets will have to support 201 times as many fleets and garrisons as the number of systems it garrisons.
And if Y equals 1,000 times Z, X will equal 2,001 times Z, so the system sending the garrisons and fleets will have to support 2,001 times as many fleets and garrisons as the number of systems it garrisons.
So for this system to work methods of equalizing the personnel and resources that the various levels of governmental star systems will have to devote to keeping their immediate subordinates garrisoned will have to be found.
6a) ships from farther away travel faster than ships from nearer, in proportion so that trip times turn out to be about the same.
If the ships can accelerate for half the journey and then decelerate for half the journey, for example, that would be the case.
On a scale much smaller than the thickness of the galactic disc, the galaxy will be three dimensional. Thus doubling the dimensions of a sphere or cube of space will increase the volume of that space, and the number of stars within it, by eight. So perhaps star systems are grouped by eights, sixty fours, five hundred and twelves, four thousand and ninety fours, etc., etc.
And the star system that rules a group of eight stars might be permitted to send fleets and garrisons to its subordinate systems at an average speed of one percent of the speed of light. Thus its fleets will spend an average of one hundred years times the distance in light years in each one way voyage.
And the star system that rules a group of sixty four stars might be permitted to send fleets and garrisons to its subordinate systems at an average speed of two percent of the speed of light. Thus its fleets with spend an average of fifty years times the distance in light years in each one way voyage, and since the average distance will be about twice as far, will spend about the same time on average. Thus they will need to support about as many fleets and garrisons.
And the star system that rules a group of five hundred and twelve stars might be permitted to send fleets and garrisons to its subordinate systems at an average speed of four percent of the speed of light. Thus its fleets will spend an average of twenty five years times the distance in light years in each one way voyage, and since the average distance will be about twice as far, will spend about the same time on average. Thus they will need to support about as many fleets and garrisons.
Assuming that the thickness of the galactic disc is 1,000 light years, a cube of space as thick as the galactic disc would be 1,000 light years on a side and would contain 1,000,000,000 cubic light years of space. There are estimated to be about 0.004 stars per cubic light year in the neighborhood of the Sun, so there would be about 4,000,000 stars in a cube 1,000 light years in a side.
It would contain eight smaller cubes with about 500,000 stars in each, sixty four cubes with about 62,500 stars in each, five hundred and twelve cubes with about 7,812.5 stars in each, Four thousand and ninety four cubes with about 976.5625 stars in each, thirty two thousand and sixty eight cubes with about 122.07031 stars in each, and two hundred sixty two thousand one hundred and forty four cubes with about 15.258788 stars in each.
Then four of the cubes with sides of 1,000 light years could make a rectangular volume with sides of 2,000 by 2,000 by 1,000 light years, containing about 16,000,000 stars; four of those volumes could make a higher volume with sides of 4,000 by 4,000 by 1,000 light years containing about 64,000,000 stars; four of those higher volumes could make a yet higher volume with sides of 8,000 by 8,000256,000,000 by 1,000 light years with about 256,000,000 stars; four of those yet larger volumes could make a still larger volume with sides of 16,000 by 16,000 by 1,000 light years containing about 1,024,000,000 stars; four of those still larger volumes could make an even larger volume with sides of 32,000 by 32,000 by 1,000 light years containing about 4,096,000,000 stars; four of those even larger volumes could make a vaster volume with sides of 64,000 by 64,000 by 1,000 light years containing about 16,381,000,000 stars, and four of those vaster volumes could make a galactic realm with sides of 128,000 by 128,000 by 1,000 light years containing 65,536,000,000 stars.
So by doubling the dimensions of each successive level of administration, there could be about 15 levels of administration above the level of single star systems.
If the galactic capital system sent out garrison fleets with speeds of 99.99 times the speed of light, the next level down could send out garrison fleets with speeds of about 0.50 the speed of light, the second level down could send out garrison fleets with speeds of about 0.25 the speed of light, the third level with speeds of about 0.125 the speed of light, the fourth level with speeds of about 0.0625 the speed of light, the fifth level with speeds of about 0.03125 light speed (abbreviated c), the sixth level with speeds about 0.015625 c, the seventh level with speeds of 0.0078125 c, the eighth level with speeds of about 0.0039062 c, the ninth level with speeds of about 0.0019531 c, the tenth level with speeds of 0.0009765 c, the tenth level with speeds of 0.0004882 c, the eleventh level with speeds of about 0.0002441 c, the twelvth level with speeds of 0.000122 c, the thirteenth level with speeds of 0.000061 c, the fourteenth level with speeds of 0.0000305 c, and the 15th level with speeds of 0.0000152 c.
If a capital of a 15th level down sector had to send ships up to 10 light years away at a speed of 0.0000152 c, the one way trip would last 657,894.73 years. If the galactic capital system had to send ships up to 64,000 light years away at 0.9999 of c, the one way trip would last 64,006.4 years. So that would actually be an advantage of ten times for the galactic capital system as compared to the lowest level sector capital.
6b) Have the population live longer the higher the level of the world they live in. Humans willing to undergo only the slightest body modifications, because of their desire to remain almost totally normal, might have lives extended to about 500 years, for example, and would live in the vast majority of star systems.
Those rare humans willing to be modified somewhat more might live for about 1,000 years, and they would tend to congregate in the systems that were capitals of the first administrative level above sing star systems. And the navy personnel send to garrison star systems in that small administrative unit would be recruited from them.
Those even rarer humans willing to be modified somewhat more than that might live for about 2,000 years, and they would tend to congregate in the systems that were capitals of the second administrative level above single star systems. And the navy personnel send to garrison the capital star systems for first level administrative units would be recruited from them.
And so on and so on, with rarer and rarer humans willing to undergo more and more extensive modifications and to become less and less human or normal for the sake of more and more extended lives, and living in the capitals of higher and higher levels of administration, and being sent on longer and longer voyages to garrison lower level worlds.
6c) Increase the tax revenue of higher level worlds.
If light speed communications and slower than light travel is the rule, there should be very little interstellar trade, and very little interstellar taxation. But maybe there is some degree of interstellar taxation.
first level: Perhaps each star system collects 99 percent of its taxes for itself, and sends 1 percent of its tax revenue up to the capital star system of the administrative level above it.
second level: That capital system collects the same amount of taxes as any other system, and keeps 99 percent of it, and also keeps 99 percent of the taxes sent from 100 other worlds. Thus it receives a total of 200 percent as much tax revenue as an average star system, and keeps 99 percent of it and sends the equivalent of 2 percent of the tax revenues of an average world to the next higher system.
Third level: The next higher star system (above 10,000 star systems) might receive the equivalent of 2 percent of the tax revenues of an average star system from each of 100 lower system for a total of 200 percent, as well as collecting 100 percent of the taxes of an average star system from itself, for a grand total of 300 percent of the tax revenues of an average star system. It sends one percent - or 3 percent of the taxes of an average star system- to the system above it.
Fourth level: The system above it (which is above 1,000,000 star systems) collects 3 percent of the taxes of an average star system from each of 100 lower systems, plus 100 percent taxes from itself. So it receives 400 percent of the tax revenues of an average system and send 1 percent of that, or 4 percent of an average system revenue, to the next higher system.
Fifth level: The next higher system (which is above 100,000,000 star systems) collects 4 percent of the taxes of an average star system from each of 100 lower systems, plus 100 percent taxes from itself, for a grand total of 500 percent of an average system's taxes, and sends 1 percent of it, or 5 percent of an average system's taxes to the next higher system.
Sixth level: The next higher system (which is above 10,000,000,000 star systems) collects 5 percent of the taxes of an average star system from each of 100 lower systems, plus 100 percent taxes from itself, for a grand total of 600 percent of an average system's taxes, and sends 1 percent of it, or 6 percent of an average system's taxes to the next higher system.
Seventh level: The next higher system (which is above 1,000,000,000,000 star systems)is the galactic capital and collects 6 percent of the taxes of an average star system from each of 100 lower systems, plus 100 percent taxes from itself, for a grand total of 600 percent of an average system's taxes, and keeps it all.
Actually there are only supposed to be about 200,000,000,000 to 400,000,000,000 stars in the Milky Way Galaxy. There are many ways to modify that arrangement so that the capital system of the galaxy receives much more tax revenue than that.
For example, if 0.0001 of every system's taxes goes to the galactic capital system, it would receive at least 20,000,000 times the taxes of an average star system; if another 0.0001 of every system's taxes is divided among the next level capitals below the galactic capital, they would each get 20,000,000 times the taxes of an average system, divided by their number; if a third 0.0001 was divided among the third level capitals each would get 20,000,000 times the the taxes of an average system, divided by their number; and so on.
If the top ten levels of capital systems each receive and distribute among themselves 0.0001 of the total galactic tax revenues each year, that comes to 0.001, a tenth of a percent, of the total galactic tax revenues, equal to the total tax revenues of 200,000,000 average systems. That should help compensate for the higher level capital systems having to send their garrisons farther away for longer periods and thus having to have more garrisons in transit at any one time.
6d) Capital systems are selected for the potential to have larger populations, wealth, and resources than ordinary star systems. And if the science is advanced enough millions of years in the future the capital systems can be rebuilt to support larger populations, wealth, and resources than ordinary star systems.
Seek out and read Larry Niven's 1974 article "Bigger Than Worlds", which has been reprinted many times.
And go to the PlanetPlanet site and read the section "Ultimate Solar System" where solar systems with more and more habitable planets are imagined. The author claims that all these solar systems are possible, that they could keep on existing if they were formed, but the more habitable planets they have the less likely they are to com into existence naturally, and the rarer and rarer such systems would be if forming naturally, instead of being created by planetary engineering on a vast scale.
The more spectacular systems are described as "engineered", since it seems statistically impossible for such extreme systems to form naturally and they could only exist if created by incredibly vast works of cosmic engineering.
Thus the higher and higher capital systems could be selected and even engineered to have more and more resources to keep fleets and garrisons in the dependent systems.
6e) The people of the capital planets believe that the ultimate horror is an independent government capable of making war on others, and thus are intensely determined to do whatever it takes to keep their subordinate systems from revolting.
6f) A combination of some of the above.
So these are my thoughts on how to hold a galactic government together with light speed communication and slower than light travel.