Several years ago at one of our weekend matches the concern of "overheating" barrels came up. I returned the next day with a Fluke digital thermometer and ten feet of thermo couple wire. The bare leades were placed four inches ahead of the breech face on the outer surface of the barrel. Right about where the rifling begins. A thin one inch square of brass shim stock was placed over the leades and electrical taped in place.
Ambient temperature that day was 90 F. From ambient we recorded the temperature increase for each shot fired. The time between shots was kept at 20 second intervals. Each shot fired elevated the surface temp by about 4-5 degrees. At shot number 10 the surface temp was at 142 degrees F....too hot to the touch of human flesh for very long. We fired another five shots with a final surface temp reaching 157 degrees. Total shots fired for this test was fifteen. Another five shots reached 159 but would not go higher. It seems we were at a point of no elevated temps beyond 159. The barrels returned to ambient temperature in 12-14 minutes exposed to a light breeze.
The rifle was a 24" stainless sporter contour in 7mm-08, charge was 38.0 grs. Varget pushing a 130 SMK.
We repeated this test with very close results with a .260 and 6.5 X 55 rifle and same barrel dimensions.
With the melting temperatures of alloy steels approaching 2600 F, I do not concern myself with hot barrels.
Your morning cup of coffee is served between 160-170 F