Years back, makers tried different things to radiate heat off from the barrel, Look at the fins on a 1928 Thompson. Then they realized that they were better off with a solid, heavier barrel- it serves as a heat sink. The coils on the heat pump are having heat delivered to them continuously.
Just HOW hot is going to be driven by powder used, loading, and weight of barrel, ambient temps, and rate of fire. I can shoot 22 LR til the cows come home, and have a barely warm barrel. Also have a very lightweight .308 that after 5 rounds rapid fire you do NOT want to grab the barrel with a bare hand.