There are a couple of sources I've found (here and here) that state that the temperature of a cave system on Earth will generally stabilize at the average surface temperature of the cave's opening.
Diving deeper, Earth has a geothermal gradient so that, "Away from tectonic plate boundaries, it is about 25–30 °C/km (72-87 °F/mi) of depth near the surface in most of the world.". So, the bottom of a 1 mile deep shaft will be about 80 °F warmer than the top, very roughly speaking. At about the 250 mile mark, it starts getting warmer much faster. It also gets warmer faster at tectonic plate boundaries, near volcanoes, etc., due to the proximity of magma.
So, for small to moderate (in D&D terms) cave systems, whatever the average temperature is will dominate the interior of the cave system. Sections nearer the entrance will have more varied temperatures, being closer to the current temperature, but only the closest area or two. Extremely deep caves will get quite warm, but it'll take a while.
Of course, D&D doesn't explicitly have plate tectonics, and the plane you're on might not contain spherical planets that are being kept warm by the physical processes that exist on Earth, but are kept warm by magic or the gods. And, of course, many worlds have an "Underdark", which my experience says is rarely outside of the "cold/moderate/warm" bands. YMMV, of course.
And, you said "underground complex", not "cave": there are plenty of examples of non-humans on Earth building climate-controlled habitats, plus D&D/PF have magic in the world, so the denizens could use that to help climate-control their house.
In my D&D/PF experience, caves are generally "Moderate" unless there's a plot reason for them to be otherwise. That allows for a lot of hints to be thrown at the players (eg., a rise in temperature hinting that a portal to the Elemental Plane of Fire is nearby).
The OP changed a bit after I posted this; to address the balance portion of the question...
With a (relevant) Session Zero
If there was a Session Zero during which the environment and climate of the adventure's primary locations was discussed, stick with whatever was decided there; if the player optimized their character for the agreed-upon setting, that's Session Zero working as intended.
Without a (relevant) Session Zero
If there wasn't a Session Zero during which the environment and climate of the adventure's primary locations was discussed, this GM would advise treating the Underdark more-or-less like the surface world in terms of climate: mostly, it's probably in the "Moderate" or "Warm" bands, since that's where most of the "civilized" races like their climate to be. Or, rather, the fact that most "civilized" races evolved or were placed into those bands is why they're "Moderate" and "Warm"; Luke Skywalker doesn't become the hero because the camera follows him, the camera follows him because he becomes the hero.
There are probably warmer parts of the Underdark, due to portals to the Elemental Plane of Fire, magma pockets, pure depth, or just the preferences of the local denizens. Similarly, there are probably colder pockets: portals to icy planes, white dragons making a home for themselves, etc..
Therefore, much like the surface, this GM would default most areas of the Underdark to "Moderate" or "Warm" with pockets of regionally-appropriate variances from that middle. I might use other weather-ey cues to add some variety without changing the actual temperature, though: is the air stale or fresh (and, if fresh, have some reason as to why); how's the humidity (high humidity in cooler temperatures might feel cold and clammy, especially to surface-adapted races); what light is available; etc..
This GM wouldn't start with any concerns about one character's ability to change the weather, especially with a 4-feat investment in being able to do so: there are a lot of other things that can happen with 4 feats which, depending on the character build, can kill things a lot more quickly than an extra damage per die or an extra caster level. Now, that said, this GM would pay attention to the fact that creatures that typically live in colder areas often have some resistance to cold. I wouldn't try to negate the feat investment by (eg.) giving everything in the Underdark cold immunity, nor would I force the weather past "Warm" terribly often, but I would definitely keep some of those options in my back pocket.
In addition, since the player has invested in making the weather important, this GM would follow suit and actually use the exposure rules (did they pick up some Cold Weather Gear?). Again, this isn't to try to negate the feat investment, but to take the cue the player has offered that "I want the weather to matter".