Differentials will get to 150 F within a few seconds of running--maybe 30. It's amazing how fast they heat up. Running temperatures are 200 F or so on the street and 220 F on the track is not uncommon. 240 F is too hot and it's time to get a cooler going.
250 F is time to stop and change the
oil before you spend lots of $$$ to fix it.
David
