Theoretical postulating aside, APUs can run for a VERY long time. It's wasteful and loud- you're much better off to pull up a newer, quiet GPU cart and plug her in. But the APU is certainly up to the task of endless hours of thankless work.
Actually, APUs DO
run at full RPM. Their minimum RPM is 100% N1 (which is the only N they have!), and aren't considered to be "over-revved" until about 110%, depending on the model.
If the APU gets down to 95% N1, all loads are shed in an effort to save the APU. Sometimes it works, and sometimes you're doomed.
Now, with EGT, you're certainly right. If all you're doing is running the generator, the EGT generally stays at a low 325-350 (on the GTCP85, I'm talking about... I'm sure others are similar). But if you open up pneumatics, the EGT can climb quickly and get ugly.
Newer APUs are MUCH more reliable than the older ones. The older ones had to be changed frequently, and are placarded INOP half the time! But newer designs have addressed issues, and with improved materials, we can see APUs with high numbers of hours between being removed from the airplane.