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.
Jim makes a good point alot of factors can vary the equation. Without getting into alot of scenarios IMHO generally speaking if it starts to feel warm to the touch it's time to let it cool down. In my 300 rum it's about 3-5 depending on outdoor temps and it's time to let it cool down but on my M1a I'll shoot the entire mag and then let cool.
Usually by the third shot in any of my mags, the barrel starts feeling warm and I usually wait a couple of minutes between the first, second and third shots. If it starts feeling warm to the touch, I let it cool until it feels same as ambient temp. Most ot the time I just wait until the barrel feels ambient between every shot.