Here's my thinking, something is "repairing" itself as it cools off and the HiPres oil system, and it's controls, would be the first place I'd look into. Either the oil gets hot, and thins, causing a drop in pressure (because it's only working marginally now) to the point where it's below the threshold for running. Or it's old oil that's foaming-up and causing the drop (the reason I ask about the oils condition).
Neither seems to be true here, so the pump could be marginal because of its age or the pressure is being bleed-off as stuff gets hot. But either of these would start to happen gradually. You'd notice problems on long, hot trips first. Did this start happening all of a sudden?
If so, that could indicate one of the HiPres oil sensors failing (ICP or IPR). These sensors are expensive so, you don't want to just "try" changing them
The ICP is pretty easy to test. Next time it won't restart, just unplug it. The PCM will see no signal coming from it and set a default value for the IPR to use as a "limp-in". An indication of a failing ICP is that, as the internal seals fail, they leak engine oil into their electrical plug so, look for sings.
The IPR has no free test like that though. What can happen with them is that the "tin nut", that holds the coil on to their shaft, can vibrate off and that coil moves around causing problems with its readings. Also, the plunger inside can get crap in it and stick (you can try "rapping" on it with a screwdriver handle sometimes). It is also possible to, carefully take it apart and inspect. It not uncommon for a failing IPR to do what you describe.
Both these sensors have moving parts that heating up can cause to bind. The very Best way to check the function of both these sensors is with the proper scanner. It would quickly tell you if they were doing their job or not. Unfortunately, you need a professional scanner or an AutoEnginuity because our trucks are not, strictly, OBD-II and the parts stores "generic" scanners are of little use.