I shoot competitively as well, very often 10 shots in under 120 seconds is the pace. During the warm months the barrel is entirely too hot to touch by the end of a string. If I have ice available, I'll set a cube above the chamber or throat and run it up and down the barrel till it's at ambient-ish temp. If not ice, I'll pour water over it BUT NOT INSIDE IT. I only do this because during the warm months the match is less popular and so I don't have as much time for natural cooling to work between stages.
How hot is too hot, when your skin doesn't like holding on to it with a firm grip, it's already too hot. The result of shooting it while it's too hot is shorter barrel life (specifically throat cooking) and most guns start opening the heck out of their group size as they heat up for a suite of reasons. Competitors firing long strings quickly are also by definition NOT letting the ammo bake in the chamber, we're closing the bolt and promptly sending the round downrange as a matter of necessity. Otherwise it's hard to get 10 rounds off in 90-120 seconds on a bolt gun while actually hitting targets.