I have a .264 WSM (a wildcat .264 made by necking down 270 WSM or 300 WSM cases), and I can tell you that the barrel, a #3 (heavy sporter) contour, would be just about fire hot if I shot 6 rounds back to back. 10 minutes wouldn't cool it off, either. Now, my light .223 with Win 748 or similar spherical powder, that'd be completely doable.
The point here is just that it depends upon the specific cartridge you're shooting. My own personal rule of thumb is to not let the barrel heat beyond noticeably uncomfortable to the touch. I'm sure others would say that's abuse, and yet others would say that's way too conservative.