It depends but It can get very hot, it depends on whether it's new gears and bearings or one with over, say ~5K miles or so (Broken-in). About 194F to over 370F.
I forgot where I stole this but it's likey from Amsoil or some other synthetic oil maker. I was shocked to see how hot an axle can get so I kept this:
A light duty truck towing 14,000 pounds was driven 500 miles. The test was conducted with both a new axle and a broken-in axle. Over level ground towing, oil temperature was measured at 230 degrees F in the new axle and 203 degrees F in the broken-in axle.
Oil temperature during a 6 percent grade the new axle was operating at 350 degrees F and the broken-in axle was operating at 300 degrees F.
Laboratory dynamometer test results simulating a truck hauling a trailer provided similar results, with level ground towing temperatures recorded at 266 degrees F with the new axle and 194 degrees F with the broken-in axle and towing temperatures (at a 3.5 percent grade) recorded at 370 degrees F with the new axle and 295 degrees F with the broken-in axle.
For actual #s. So far I only see up to 180 on my 98 5.4 with 4.56 gears. bout 165 up to 65. 180 above that. rain brings it down to 140. Will be interesting to see if/when I do my first hot weather heavy trailer pull. Sure don't want to see that 300 deg mark.