1991 Corolla. Will not start after engine is hot

Hello,

I have a 91 Corolla with 161K miles on it. In the mornings when I go out, it starts and will run as long as I need it. I usually drive 40+ miles to work each day with no problem. When I get to work, I let it set for several hours, it will start up and run fine again, but if I have to stop for some reason and shut it off, it will not start unless I let it sit for 2 to 3 hours.
When I say it won't start, I mean, there is nothing when you turn the key, the starter doesn't even engage. No clicking, nothing. The radio, fans, lights etc all work and come on. Just nothing at the starter.

1991 Corolla
161,000 miles
Automatic

Any help would be appreciated.

Thanks