No it doesn't "start to cool earlier" . Both barrels start to lose energy (heat) as soon as the source of the energy (the heat and pressure of the cartridge firing and friction) is removed.
wrong.... lets say we go to the range..... I start firing off my mini 14 as fast as I can.... after ten shots you reach over and wrap your hand around the barrel while I am still firing.... do you think it is going to be cold?
No, of course not because air is being dissipated from the barrel surface as yo uare firing.... it is both heating and cooling at the same time... only reason it heats up is because while you are firing the heating force will always outdo the cooling force... (The gun heats up quicker than the barrel can cool it down)..... doesn't mean it isn't cooling.....