I had a PRI carbon fiber tube on one of my AR's and can tell you that they do get HOT if you leave them out in the sun for a while! They do have cooling vents in the tube to dissipate barrel heat unlike the Hogue or Bushmaster Vmatch tubes.I have a carbon tube on one and a standard al freefloat tube on the other
the carbon gets warm but not hot, though I haven't put as many rounds as quickly through it (6.8 vs 5.56)
I have one of the Troy Extremes coming for my new 6.8 upper and assuming I like it may get one for my 5.56 as well as it gets more extended shooting sessions through it