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Although I don’t have any supporting data, I would say the difference in differential type and the heat generated would be minimal at best. The clutches in a track loc, or the helical gears in a Torsen are only turning when there is a difference in the rotational speed of the tires. The majority of the heat comes from the pinion bearings, then carrier bearings.
The gear ratio would have more influence on heat as a lower ratio. 4:10 would run hotter than a 3:31 because the pinion would be turning faster at the same speed. That being said, I would pick the gear ratio for the tracks you run. I’m a TT guy, so axle heat in my 2014 isn’t a problem, so maybe some of our endurance racers can chime in.
Yes lots of gears churning through the oil in a torsen so they do run a bit hotter. As stated above though final gear ratio makes a bigger difference. I just did 4.09’s and there is no way you can/should do 20min sessions without a diff cooler.