Re: Barrel Heat
If the cartridge is the same as you used before, one reason the new barrel heats up faster may be due a smaller bore/groove diameter. Or if the new barrel has a much faster twist. If it's rougher than the old one, that can cause a barrel to heat up faster, but usually only during the first few shots.
Some tests were run years ago comparing four different barrels of the same material, had the same countour ('cept for the fluted one), bore and groove diameters and were chambered for the same cartridge. The differences were one was shiny smooth, another fluted but a bit larger in diameter to have the same weight as the others, one was fine bead blasted and the last was coarse bead blasted.
Each one heated up a bit faster or slower than another and cooled off the same way. Evidence showed the following:
Shiny smooth barrel heated the fastest and cooled the slowest.
Fine bead blasted one was second in heating and third in cooling.
Fluted barrel was third in heating and in second place cooling.
Coarse bead blasted heated the slowest and cooled the fastest.
If you want your new barrel to cool faster, I'd get it coarse bead (or sand) blasted. Fluting it will make it less stiff plus changing the bore and groove diameters; more at the ends of the flutes than between them. If it's a button-rifled barrel, the area between the flute ends will be a bit smaller in diameter with an oversize ring at each end. If it's a hammer forged barrel the dimensions between and at the flute ends will reverse. Hammer forged and button rifled finished barrels typically aren't as accurate after fluting. Bore and groove dimensions won't change much at all if it's a cut-rifled barrel and accuracy usually doesn't change if light fluting cuts are made.
I don't know if chrome moly heats and cools the same as stainless steel or not. Do an internet search for steel properties, type 4140 chrome moly and 416R stainless; that may give some numbers to help out.