I would only worry about the heat if I was shooting a flurry with black powder. I could see that leading to a popped barrel rib.

I once watched a fellow shoot a flurry for 2 minutes with a pair of guns at the Vintage Cup. With black powder. He was a good shot and it was very impressive but I bet those guns were hot!

I think pairs and triples for driven birds weren't all for rate of fire. It may have prevented guns getting too hot in the black powder days.