Going underground is not a solution to rising temperatures. The reason why it’s good at keeping its temperature somewhat constant is because it is a poor heat conductor.
It’s fine for a cellar or something like that but larger structures still require air conditioning. See for example the London Underground where they have issues with the deep underground lines heating up over the years.
A major problem with the London Underground is the thermal energy introduced by trains and support equipment itself.
Spaces inhabited just by people and incidental devices (e.g., LED lighting, fans, personal electronics) should see far less heating effect, particularly if they're well-ventilated such that any accumulated heat could transfer out quickly.
Houses have had cellars and basements for centuries if not longer, and those aren't known for getting uncomfortably warm.
For large enough subterranean structures, air exchange becomes a challenge.
One place this is evident is with naturally-formed caves, where changes in atmospheric air pressure translate to air movements in and out of the caverns themselves, one consequence of which is the many "moaning caverns" --- the airflow and resonances are sufficient to create audible sounds associated with the movement. It can take hours for pressure to equalise if the volumes and apatures are sufficiently divergent.
Warm air will rise, and cold sink, yes, but if there are any trap-type structures (e.g., a tunnel which rises then descends again) the rising/falling air will be trapped at a local maximum or minimum. This is directly comparable to a thermal inversion associated with smog and urban pollution, often occurring during winter time. The only way around this is forced or active ventilation.
You've also got the "stale air" problem. If there's only a single entrance, or entrances are only on one part of the cave complex, air as a whole moves in and out of the caverns, but the deepest and furthest recesses see relatively little interchange. This leads to one of the hazards of spelunking in the event noxious gasses accumulate or oxygen is displaced.
For the London Underground (and other subway systems), much of the air exchange is provided by the trains themselves, through the "piston effect" of a trainset moving through a low-clearance tunnel. You'll experience this as the gust of wind which preceeds a train's arrival, or the suction as it leaves a platform. This effects some thermal transfer, but mechanical ventilation is still required.
Non-transport-oriented underground or partially-subgrade structures typically have far more relative air transfer, and would generally have less-signficant airflow / air exchange challenges, though I suspect there's still quite a bit of HVAC engineering involved, for thermal and other reasons (stale room-space air, radon emissions, mould & moisture, etc.).
It’s fine for a cellar or something like that but larger structures still require air conditioning. See for example the London Underground where they have issues with the deep underground lines heating up over the years.