If the CF barrels do not run cooler and do not shed heat faster than a steel barrel, then besides manufacturer misrepresentation on this aspect, CF barrel owners could think their barrels will cool faster than a steel barrel and be more immune to rapid rates of fire. Owner's may feel comfortable with higher rates of fire. I believe the steel cores of CR barrels will actually run hotter.
On top of misrepresentation, this would be a complete disservice to CF barrel owners.
I posted in another thread and solicited by open invitation, that anyone desiring to correct my claims of less efficient heat transfer was invited to participate in the thread and support their claim of better heat dissipation with CF barrels. Even manufacturers could sound in. No takers for a week or so now. The truth would be best fleshed out through testing of their CF wrap for thermal conductivity (independent test to ensure credibility), and then providing those numbers/values. If the numbers are different in the longitudinal direction than the radial direction, then provide a thermal conductivity value in both directions. Virtually all CF barrel makers claim better cooling, yet none that I've seen provide this documentation. This would be no different than insulation manufacturers providing the R-value of the insulation products. Higher R-value = less thermal conduction (better insulation rating) per unit thickness of the insulation media.
Shy of that method, a guy could test two barrels of equal contour, one CF and the other steel, at equal rates of fire with the same cartridge chambering. Could hook up some temperature sensors and register temps at multilple locations along the exterior of the barrels for a digital readout. This method presents more variables besides strictly thermal conductivity values, but it could provide good data.
Next cruder method is the end user can just feel by hand the temperature difference at the location the CF barrel manufacturer claims their CF barrels will shed the bulk of the heat of fire most effectively. If it's longitudinally, then feel the steel at each end of the CF wrap. It should be notably hotter than a plain steel barrel of similar contour at the same location and rate of fire - with the same ammo. If the claim is radial heat loss, then the carbon fiber wrap should be warmer to the touch at all external surfaces, compared to a plain steel barrel of equal diameter.
If you feel no difference, the conclusion should be straightforward and obvious. If the CF wrap is cooler to touch, as one owner claimed to support his belief that his CF barrel cooled more efficiently than steel, then it's obviously horse-pucky.
If I owned a CF barrel, I would be more cautious with high rates of fire than with a plain steel barrel. Certainly so until the CF barrel manufacturer presented the thermal conductivity values of their CF wrap, and I could confirm that their CF wrap dispersed heat more quickly than steel. Even then I'd still be doing the touch test for final confirmation of higher exterior surface temps during some repetitive fire, compared to comparable contour steel barrels.