Yes and no, a light and heavy barrel subjected to the same heat input.
lets say light gets to 300dc and heavy gets to 100dc. The light will lose more heat to atmosphere even allowing for less surface area simply due to temperature differential to atmospheric temp.
Heat will conduct far quicker threw steel than air, touch a red hot bit of steel 820dc or wave your hand threw a lpg oxy flame 1200dc.
A heaver barrel will take longer to heat and cool down than a light, as a barrel heats the temperature differential between the flame front in your precious throat and barrel temp gets less and so does the rate of cooling of the throat.
If what you are worried about is the temp and subsequent erosion of the throat, then a heavy barrel will keep that throat cooler for longer of your first string of rapid fire. then it will take forever to cool.
Simple physics
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