93 Crown Vic will not start when hot

Have a 1993 crown vic. Starts fine when cold. After it has been running for a while and is up to temp. If shut off, when you go to restart it, acts like the battery is dead. Did a load test on the battery and it was fine. Plus do not lose any pre-programmed radio stations. So that leads me to believe problem is not with the battery. Let the car sit for awhile and cool, it starts right up. Any help is greatly appreciated.

Could be the starter, a very common problem. Car starts fine cold, but won't turn over when hot.