Given an Earth-like planet, what sort of geological features or weather systems are required for a place to have the same average day and night temperatures throughout the whole year? Is such a thing even possible?
The closest I could think of were tropical climates that are hot all year long (though even these have cooler wet seasons), but I'm looking for something more temperate, around 15-25°C during the day.
The minimum area of the region should be roughly as big as Belgium.
A large body of water can have an enormously stabilizing effect on the weather. It is a common joke that weather at San Diego is the same year round.
While not entirely true, it does have some truth to it. Summer in San Diego usually has a range from 65F to 75F, while the Winter range is 50F to 65F. I live at about the same latitude, but several hundred miles inland (and several thousand feet higher) away from any body of water and temperature can range from 63F to 92F in summer and 27F to 53F in winter.
Water does not guarantee a mild climate though, as the Great Lakes region is known for its Lake Effect snow which can drop a substantial amount of precipitation in a short time.
Seasons are pretty universal, being caused by the angle at which the sun's light hits the planet. To make a point on a planet without seasons, you have two main options:
A planet with an extremely thick atmosphere that blocks out the sun and heats the planet through greenhouse gases will have less temperature variation over the entire planet. Of course, such a planet will not actually be Earth-like.
A large body of water is part of the answer.
The water circulation in the Pacific Basin is clockwise (bringing cold water from Alaska to California) in the summer, and counter-clockwise (bringing warm water from the tropics to California) in the winter. The average water temperature off San Francisco (and probably also San Diego) is higher in the winter than in the summer.
This has a lot to do with the nearly constant temperature in San Diego.
The reason of why we have seasons, and those are always opposed in the North and South hemispheres, is because the axial tilt in the Earth rotation axis, which is 23.4°. This gives a huge difference of the Sun's light incidence over the planet between the Summer and Winter seasons.
Let's calculate the higher and lower effective Sun's light incidence over various regions of the Earth using a simple formula: c = cos(incidence angle)
In the regions near the 0° latitude (the Equator) the difference is from 100% Sun's light incidence at the equinoxes and 92% at the solstices (both twice a year) which gives the tropical regions barely the same temperature every day.
At higher latitudes (let's say Rome, at 41° 54′ N) the differences can be from 95% at Summer solstice to 42% at Winter solstice, which makes a hot Summer and a cold Winter.
If you want a region to have the same temperatures every day a year, your world should have its rotation axis perfectly perpendicular to its orbit (0° axial tilt). This way the temperature in a certain region will depend only on its latitude since you will no longer have seasons.
The simplest answer was given above: give the planet a perfectly circular orbit with zero obliquity (no axial tilt) and each part of the planet's surface will receive the same energy from the star year-round. You could take this a step farther and assume your planet is orbiting a red dwarf star such that its habitable zone is close enough that the planet becomes tidally locked to the star, so the planet always shows the same face to the star. This can make the planet look like an eyeball (see here: https://planetplanet.net/2014/10/07/real-life-sci-fi-world-2-the-hot-eyeball-planet/ or here: http://nautil.us/blog/forget-earth_likewell-first-find-aliens-on-eyeball-planets). For an eyeball planet the temperature structure would vary a lot across the planet but not in time.
The thicker the atmosphere, the more uniform the temperature across the planet's surface (Venus is the extreme example, but one to be avoided if you want life on your planet). I don't have a good feeling for the effect of the spin rate on the stability of the local temperature. Faster-spinning planets will have more "banding" structure in their atmospheres but I don't know if that is a stabilizing force. Someone with a specialty in atmospheric dynamics would probably know.
While San Diego has remarkably stable temperatures year round, there are other places that are even more stable. In tropical highlands, places get the same stable insolation as the steamy lowland tropics, but the air temperatures are mitigaged by altitude. This results in basically the most pleasant places on earth.
There are several tropical highlands straddling the equator. Africa has the Rift Valley mountains, from Ethiopia and Nairobi in the Kenyan Highlands through Rwanda and Burundi. All the cities in the Andes from Caracas to La Paz have pretty stable temperatures. And through extraordinarily wet, the highlands of New Guinea have very stable temperatures too.
It's called a cave. Go down about a hundred feet and the weather pattern is very stable. It's a bit chilly, but not impossible to make a comfy place.
Avernum takes place in a giant interconnected cave system.
The same could be said for the Underdark, though I've only been there in Neverwinter Nights so I can't say if there are storms.
Journey to the Center of the World also featured interconnected cave systems.
If you want something a little more open, a Dyson Sphere or Ring should also have the same temperament everywhere unless something's gone terribly wrong. Like the heat source drifting closer to one wall (floor?) than another.
Planets can't really be temperament unless they're tidal locked to their star. Then it's all about placement. The side that's locked toward the star would be hot, then a safe zone around the Dawn Belt and finally a frozen wasteland on the other side. Our moon is Tidal Locked to the Earth meaning that for millions of years, we've only seen one side of her.
If you have a dimensional disturbance story (a.k.a magic) or a possible Fey Loci then that would solve the problem as well.
This may not be the answer you are looking for, but thought I'd post it anyway: if a country is close to the equator (e.g. Sri Lanka), the temperature would be same (approx. 30 degrees C) all year around. The day time is also fairly constant i.e. sun rises at about 6am and sets at about 6pm (variation through the year may be about 1 hour).
