Let's play "what part is failing?"

[vsp]

The apartment was at a normal temperature, around seventy.

If it was an overheating problem, I would think that it wouldn't stay on for any length of time at all. Once this gets over the hump, so to speak, it'll stay on for days. I'd just leave it on and plugged in, but it's thunderstorm season and I'd rather not tempt fate even a little.

If it was a car, I'd get the starter replaced, but computers don't have those (that I know of).

My next step in the meantime is to get some compressed air and clean the hell out of the innards, particularly the fans.