If the carbon wrap cools the barrel better, then the exterior of the barrel should warm to the touch more quickly. Because heat from the bore would transfer thru the steel barrel, and then thru the carbon wrapping, at a higher/faster rate.
I've not invested the time to research the claims that the carbon wraps cool the barrel faster than a simple steel barrel. Because I don't own any carbon wrapped barrels.
I will say I remain WAY skeptical of those claims.
Fire 100 high pressure loads at a steady rate over a short period of time and the barrel should be HOT, if the barrel transfers heat from inside to outside more quickly than a plain-Jane steel barrel.
Makes no sense whatsoever, to anyone with some education in the principles of heat transfer.