I had a corvette once with the same problem after replacing a starter. It seems the new starter developed a condition known as heat-sink. this would cause the starter to not engage because the starter was hot. I've seen mechanics tap the starter with a wrench when this condition occurs. Sometimes it works, sometimes not. I'll bet you any amount of money it's your starter "heat sinking" as they call it. A replacement starter may be what you need. People usually don't think of this when this type of thing happens. They start barking up a wrong tree tearing everything apart to find the problem. Since it starts when it's cooled off a bit, i'll bet that's what it is.