This one will get LOTS of different answers!! Depends on so many different things---bbl. material, mass of the bbl., air temp, sun/no sun, powder type, pressure level----maybe some other things I haven't thought of.
Anyway, rate of fire is probably as important/more important than how hot the bbl. feels. In actual hunting/tactical situations the shooter doesn't get to set the rate of fire.
When doing load work, I shoot one round every 4 minutes or so until the bbl. gets warm--I can hold my hand on it without discomfort. Then I let it cool to nearly air temp is before shooting again. Also, I sometimes shoot 3 round or 5 round groups fairly quickly to simulate hunting/tactical situations.
IMHO, anytime a bbl. gets too hot to hold a hand on, it has been fired too quickly or too many times without proper cool down. Combat/full auto situations excepted.
Cheers,
Jim