The heat capacity (specific heat) or carbon and stainless steel are actually identical for all practical purposes at 500 J/kg*K...
And technically you should note a faster, more general heating in carbon steel rather than stainless in a short time period (unlikely any difference would be noted at all by human senses, but still, if one was to be noted). If we are talking firing 1 shot per 15 mins or something, maybe, possibly, I can see what you're saying, as heat conducts faster through carbon steel than stainless (51 W/m*K and 16 W/m*K respectively), perhaps you would note the difference in thermal conduction; but I doubt it. I own both carbon steel and stainless barrels (many, actually) and have not noted this trend... Most of my target rifles are stainless barreled and I haven't noted increased heat in them vs the carbon steel barreled rifles I shoot. Could be, I just haven't noted it.