Only place i heard about the temp of the barrel being a problem is machine guns. But a long range rifle, never heard anything about it. I heard a lot about cars and when drag racing they like to run hot or colder. My gun loves it hot, she likes to shoot hot and when hot smaller than 1" groups at 100 yards. And the gun is a '39 Mosin Nagant 91/30.
O.K., today I found a reference to "how hot is too hot" on a Darrell Holand shooting video. He states that you should operate your barrel between 80 and 150 degrees. He also states that much above that can shorten barrel life. That was the original question I was trying to get which is great! Now to take a thermometer out and check to see what that feels like on the old hand.