I found this neat little document of a study done by the army on M4 barrel heat. The test involved firing 280 rounds full auto in the span of a minute and a half and then letting it cool down.
Here are the two most interesting pages from it (I think).
https://i.imgur.com/RxKtopU.png
RxKtopU.png
https://i.imgur.com/ZHQug90.png
ZHQug90.png
What is interesting is that the hottest point is 4" into the barrel. I would have guessed the gas block area would be hottest.
Source: https://apps.dtic.mil/sti/pdfs/ADA285699.pdf
Document name: AD-A285 699
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