-to power themselves if they get hot enough to cause damage, so the fact that it apparently didn't power down is in your favor. It's been a couple of years since I did any serious measurements on my PCs, but when I did, I noticed that it was always hotter when I looked in the BIOS. My guess is that Power On Self Test stresses the CPU and gets it hotter than after it's finished booting. Games of course are stressful, so a lot of things enter into the equation. I'd get a temperature monitor and see how things look a few minutes after booting vs what the BIOS says. I use Speedfan but there're plenty of others as well.
Good luck.