Part One of Seven: The Problem.
In my humble opinion you can use almost any type of star as the star in the system, as long as some advanced civilization long ago more or less built all the planets in the system and seeded the ones in the habitable zone with life. And the planets in the habitable zone would have to be arranged in a ring in a shared orbit.
Others have explained how implausible your star is.
Astrophysicists were beginning to calculate the future evolution of stars at least as early as the early 1950s and 1960s.
Stephen H. Dole, in Habitable Planets for Man, 1964, discussed the qualities necessary for a world to be habitable for humans.
For example, a world has to have an atmosphere with a partial pressure of between 60 and 400 mmHg of oxygen to be breathable for humans. On pages 61 to 63 Dole discusses how long it co took Earth from its initial formation to acquire an atmosphere with enough free oxygen, produced by photosynthetic lifeforms. The present day answer is about 4 billion years, meaning that Earth has been habitable for humans for only about 600 million years.
Dole concludes that: "In general, it is probably safe to conclude that a planet must have existed for 2 or 3 billion years, under fairly steady conditions of solar radiation, before it has matured enough to be habitable."
On pages 67 to 72 Dole discusses the properties of the primary. Dole says that stars shine fairly steady on the main sequence until their available hydrogen is mostly used up, then swell into red giants, destroying life on any previously habitable planets, and then shrink to white dwarfs, perhaps having become novas during the process. More massive stars use up their hydrogen quicker and remain on the main sequence for shorter times.
Stars potentially capable of having habitable planets belong to spectral classes F, G, and K, in order of decreasing mass and luminosity. Spectral classes are further subdivided with numbers from 0 to 9 in order of decreasing mass and luminosity. Roman numerals are used to describe whether the stars are main sequence stars - V means main sequence.
According to the information available to Dole at the time of writing, stars with mass equal to or less than 1.4 solar masses and spectral types of F2V or colder, would remain on the main sequence for at least the minimum time of 3 billion years Dole considered necessary for a planet to become habitable.
Low mass dim stars have their circumstellar habitable zones (which Dole called "ecospheres"), where planets are expected to have the right temperatures for liquid water using life forms, so close to them that planets in them would become tidally locked, which Dole considered incompatible with habitability. Dole calculated that the inner part of the "ecosphere" of a star would be too close starting with a mass of 0.88 Sun and the whole ecosphere would be to close to the star starting with a mass of 0.72 Sun, a K1V class star.
So any science fiction writer who read Habitable Planets for Man in 1964 or later would realize that stars of type F2V to K1V were the only classes of stars likely to have planets with atmospheres breathable for humans. And if they wanted their stories to seem scientifically plausible to their more scientifically educated readers they would restrict depicting human habitable planets to those orbiting stars of spectral types F2V to K1V.
Scientific calculations show that a star which has gravity many times as strong as the Sun's must be many times as massive as the Sun, and thus must have a much higher surface temperature than the Sun, and so it couldn't be the same color as the Sun.
A star could have the same color as the Sun, and thus have the same surface temperature as the Sun, and at the same time be many times as luminous as the Sun, if it has a surface area many times that of the Sun, and thus had a larger radius and diameter than the Sun.
Such a star is called a subgiant, giant, supergiant, etc. depending on its size and luminosity. And the problem with having human habitable planets orbiting such a star is that giant stars are former main sequence stars which swell up into giant size at the ends of their main sequence lifetimes, and as they become much more luminous they roast any former habitable planets they had, killing all life on them. And the giant stage should not last long enough for the planets which are now getting the right amount of light from the star to become habitable for humans.
Part Two: Solution One.
However, some giant stars might have relatively stable luminosities for billions of years. So a planet suitable for life which happens to be the right position around a star might be thawed out and warm enough for life for billions of years while the star is a giant star.
But I am not an expert and can't say which mass of star would have the longest lasting habitable zone when in the giant phase, or how wide that giant phase habitable zone would be. So I don't know whether you could fit 12 planetary orbits into the habitable zone of such a star in the giant phase of its life.
Part Three: Solution Two.
Possibly an advanced civilization in the past has created a lot of habitable planets in the habitable zone of a star in the giant phase of its life. The civilization would have had to terraform planets in the new habitable zone to become habitable millions and billions of years before they would naturally become habitable. Or it would have to move habitable planets from other stars systems into orbit around the star. Or it would have to build planets in orbit around the star and then terraform them to be habitable.
That would be a vast project, but there is no scientific law which makes it impossible for a civilization to be powerful enough to complete such a project.
Part Four: Solution Three.
Or you could put the planets in orbit around several stars, whose combined luminosity would more than that of a single star. As you may remember from Part One, Dole in 1964 believed the most massive and luminous star which could remain n the main sequence for as long as three billion years would be a class F2V star with 1.4 times the mass of he Sun.
I will go with a class F3V star with 1.42 time the mass of the Sun, 1.578 times the radius, and 4.68 times the luminosity of the Sun, as the most massive star which might have habitable planets.
Since the Sun has a radius of about 695,700 kilometers, an F3V star would have a radius of 1,097,814.6 kilometers, and a diameter of 2,195,629.2 kilometers, or 1,364,300.7 miles.
So imagine two F3V stars making a binary system, orbiting around their center of mass in almost circular orbits. They should be separated by ten times the diameter of one of the stars, or about 21,956,292 Kilometers.
Now imagine another binary pair of F3V stars, with the same separation between the two stars.
Now imagine that the two binary pairs orbit around their center of mass. The two pairs have a separation ten times the separation between stars in a pair, or 100 times the diameter of one of the stars. So they are separated by 219,562,920 kilometers. That is a separation of 1.4677 times an Astronomical Unit, or AU, which is the distance between Earth and the Sun and is defined as 149,597,870.7 kilometers.
If the planets in the system all orbit around all four of the stars, the closest planets should orbit at distances of at least three to five times the separation between the two pairs of stars, in order to have long terms stable orbits. Thus the nearest planets should orbit the four stars at a distance of at least 4.403 to 7.338 AU.
Each star has a distance at which an orbiting planet would receive exactly as much radiation from the star as Earth gets from the Sun. I call that the Earth Equivalent Distance, or EED. To find the EED of a star find the ratio between the star's luminosity and the luminosity of the Sun, and then find the square root of that ratio and multiply it by 1 AU.
Since in this case the "star" is actually four stars, each with 4.68 times the luminosity of the Sun, and thus a total luminosity of 18.72 times the luminosity of the Sun, the EED will be at 4.326 AU. Thus the habitable planets in such a system would have to orbit farther from the stars than than the EED.
Part Five: More Problems.
It is fine for the habitable planets to orbit beyond the EED of the quadruple star so long as they are still within the habitable zone of the stars.
And it would be easy to adjust the inner and outer limits of the Sun's habitable zone to account for the different luminosity of a star. Except for:
This list of estimates of the inner and outer edges of the sun's habitable zone made during the last 60 years shows that some estimates a very different from some others.
So which estimates should a writer use? A writer can be certain that a planet at the EED will be in the habitable zone of its star. And the farther a planet orbits inside or outside the distance of the EED, the less certain one can be that it will be in the habitable zone.
Worse, the spacing of planetary orbits is not random. The gravitational interactions between planets prevent two planets from having orbits too close to each other. Each planet has a forbidden zone around its orbit that any smaller planet would be ejected from.
I don't know how to calculate the sizes of planetary forbidden zones. The size of a plant's forbidden zone depends on the mass of the planet, the mass of the star, the eccentricity of he planet's orbit, and the distance at which the planet orbits the star. But I don't know the formula.
The orbits of the planets in the TRAPPIST-1 system are separated by distances of hundreds of thousands of kilometers. With an average orbital separation of one million kilometers, 12 planetary orbits could be squeezed within a distance range of 11,000,000 kilometers, and most stars should have habitable zones wider than that.
But I think that the true minimum separation between planetary orbits is probably a ratio between the orbits instead of a distance in kilometers.
the smallest known ratio between the semi-major axis of planets in between Kepler-36 b and Kepler-36 c. B orbits Kepler-36 at 0.1153 AU and c orbits at 0.1283 AU. So he orbit of c is 1.11275 times as far from the star as the orbit of b.
so if the innermost habitable planet in a system orbited at distance X, the next one out would orbit at 1.11275 X, the 3rd at 1.2382 X, the 4th at 1.3778 X, the fifth at 1.533 X, the sixth at 1.706 X, the seventh at 1.898 X, the eighth at 2.112 X, the ninth at 2.350 X, the tenth at 2.615 X, the eleventh at 2.91 X, and the twelfth at 3.128 X.
So to have space for 12 planetary orbits in the habitable zone of a star, the outer edge of the habitable zone would have to be at least 3.128 times as far from the star as the inner edge. And most estimates of the Sun's habitable zone don't have that great a range of relative distance.
If only scientists could find a theoretical basis for squeezing more planetary orbits into a habitable zone.
Part Six: Solution Four.
The PlanetPlanet Blog by astrophysicist Sean Raymond does have a solution. In our solar system all eight planets orbit around the Sun in the same direction as seen from "above: the plane of the planetary orbits. Most of the moons of the planets in our solar system orbit their planets in the same direction that those planets orbit the Sun. That is called having prograde orbits. But some moons orbit around their planets in the direction opposite to the planetary orbital direction, in retrograde orbits.
In this post: https://planetplanet.net/2017/05/01/the-ultimate-retrograde-solar-system/
Raymond mentions a paper by Smith and Lissauer which shows that planetary orbits can be closer together if prograde and retrograde orbits alternate.
Raymond says that with Earth mass planets and all prograde orbits, only four planetary orbits can fit within the habitable zone of a star, but with Earth mass planets and alternating prograde and retrograde orbits eight planetary orbits can be fit within the habitable zone of a star.
This is a step in the right direction. And maybe you can be content with only eight planets in the habitable zone of your star.
But Raymond considers it impossible for such a star system to form naturally and says that it would have to have been created by a very advanced society in a tremendous feat of engineering entire worlds.
Part Seven: Solution Five.
IN the next post: https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/
Raymond finds a way to pack more than 12 planets in a star's habitable zone. At first he says that up to six planetary orbits can fit within a star's habitable zone.
Then he mentions another paper by Smith and Lissauer, which describes how a ring of planets sharing the same orbit could be long term stable. Their calculations work for 7 to 42 planets of equal mass equally spaced along he circumference of the shared orbit.
So who says that scientists never do anything nice for science fiction writers?
So you could put your 12 planets in the same orbit within the habitable zone of your star, which can be any star capable of having habitable planets. And you can put the other 56 planets in one or two shared orbits closer to the star than the habitable zone, and in several shared orbits outside the habitable zone, and maybe have a few of them in individual orbits.
And Raymond also believes that a star system with many co orbital planets like this would have to have been created by an advanced civilization instead of forming naturally. Which means the star need not be a star which would naturally have habitable planets, so long as its radiation is not too harsh.