Would that GPU really heat up that fast?
Yes absolutely.
I once spent a significant amount of time
modifying fans and airflows and such, trying to get the quietest possible computer case I could manage.
The system needed to double as a both a quiet studio recording PC as well as a 3D gaming machine. So I spent a significant amount of time stress testing the video card and monitoring its temperatures, to make sure that my fan-quieting work didn't jeopardize the GPU.
My GPU core, when running full tilt, once got as hot as 87c at its worst-case stress-test peak, although it usually never breaks 80c during games. It "idles" (i.e., surfs the web and does things that don't tax the video card) at around 60c.
And the temperature change due to the change in card activity can be surprisingly fast. It can change from that 60c to that 80c I quoted above in *seconds*.