When I drive for a while the rear end gets real hot. I can't hold my hand on the cover for no more than 3 seconds or it will burn my hand. Everything was set up correctly and I have at least 500 miles on it. Is this normal?

what gear oil did you use? Id suspect something overly tight but thats for the experts like Big Gear Head or just change the gear oil to a synthetic and the temps will fall quite a bit.

There is a break-in procedure... With only "about" 500 miles it may be broke-in by now and run cooler.
When we install new bearings on our mills and fans at work we look for this "heat spike" to show it's self letting us know it's "run-in".

I'm pretty sure that I used the best lube (or most expensive) that I could have. I'm pulling the cover tomorrow to just look at everything. I don't think it's the oil, I've used several types (all for limited slip) and everyone still gets hot.

Shoot the cover with a temp gun and see what it reads. Then, shoot the nose to read pinion brg temps. Both should be the same.

I would think any temp up to 160*F would be good. Temp of 130*F is the figure one of my mentors told me.

The threshold of pain is about 130° F, so if you can hold your hand on it for a few seconds it isn't really all that hot.