Actually, a larger diameter barrel will exhibit an increased rate of thermobuoyant convective cooling. The Biot number of a AR-15 barrel is only 3.4e-5, so the temperature of the barrel will not vary internally. I computed the heat transfer rate of a 16" barrel with a 10 and 20mm overall diameter as an example at 200 degrees C with 25 degree ambient air. The 10mm barrel has an average heat transfer of half a watt while the 20mm OD barrel transfers at 7.4966W. That presumes they are the same temperature, and the thinner one would require less shots to get up to that temp. So, a thicker OD definitely leads to faster cooling, at least until the thing is so thick that the Biot number approaches 1, which would require something several feet in diameter. I'll attach my calcs.

