Heat in an AR barrel that you will tolerate is relative to the purpose at hand. In combat, shoot away, and if it gets too hot to hold, pour water down the breech and resume.
The main damage is done in the barrel throat. As the barrel overheats, the surface of the throat starts to look like dried mud, and minute pieces start to flake off. As the throat erodes, the rifling disappears from the chamber forward. I've seen old military barrels with no rifling for 3 inches forward of the chamber but would still hold 3" at 100 yds. Disastrous loss of accuracy for a match rifle, but would still keep Charlie off your back.
So how often do you want to buy a new barrel, and what accuracy do you need? Steel that is too hot for your hand to hold is being damaged a little at each shot.