My 1989 Mustang 5.0 starts fine when the car is cold. After it is warmed up when you shut it off you have to wait at least 20 minutes before restarting or it turns over hard if you don't wait it won't start.

Probably just over due for a tune up.

Sounds like the fuel pressure regulator. if you have access to a pressure guage check the pressure when it will not start.

be looking at vacuum leaks and stuck valves when cold..

Automatic or stick ?
If its an automatic, it could be something in the torque converter or transmission. The torque converter clutch could be stuck.

TX valve freezing?

Have you replaced glow plugs lately? and does it white smoke when you drive it?

Sounds like fuel vapor lock. The fuel gets hot and changes from liquid to gas before entering the cylinders.
Look for metal fuel line bent and touching hot engine part, or no longer in its clamp. You might also insulate the fuel line with pipe insulation from Lowe's or Home Depot.

I had a reliant that did the same thing I put in a manual choke control cable and used it for another year before I got rid of it for a newer car, hope this helps

Make sure the inside of your throttle body doesn't have a gunk build up. If so, clean in out with some carb. cleaner really well. If the crack is plugged up in the throttle body, air cannot get into the engine at idle in which the car will die. Are you running an aftermarket cam? If you are, drill a 5/32 hole in the center of the lower half of the throttle plate in the throttle body. This helps a great deal as well.

Its because that the engine is very cold, try to worm it up more time before you start driving.