I personally try to back down before outside of barrel reaches 200F.
I use a barrel cooler, have used it in competition, or will set a gun/barrel aside. I've run room temp water through barrels in competition and promptly won wood. Others thought I was crazy, asking for gatoring....I dunno, I just didn't want my glue-in to let go!
I don't know of any established parameters..... I've heard things from "Duhhh! The melting point of steel is 2500 degrees, y'er not that hot!!!" To "when spit sizzles, stop" to "The barrel doesn't shoot until it's heated up" to the old standby "If you can't hold your hand on it it's too hot."
I personally stop at around 200F
My barrels do last a long time.
Hopefully someone who knows more will chime in
al