After driving my car and the engine is hot,it will not start again until the engine cools down.

All signs point strongly to a failing fuel pump
They often go out when they get hot, and begin to work again when they cool down. Telltale sign.

COULD BE A FAULTY THERMOSTAT.

check coolant temperature sensor

fuel filter inline some were

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Hi there have you tried the ignition coil as this sounds like the coil is getting hot and creating an open circuit thus no hi tension voltage to the spark plugs,once it has cooled done the car will start again until
it repeatedly gets hot and shuts down very common in pre coil pack vehicles,easy test try a spare coil and drive for same or more distant if not fault them coil is U/S

IT is most likely the thermastat but i would not suggest driving it until it fixed as that could cause the head to crack witch costs around $213 - $250

I would say your starter is faulty. Since the vehicle runs fine, and doesn't start up after being shut off until it has cooled down, it sounds more like a starter that loses contact when it is hot. You could check the starter terminals for 12 volts using a multimeter when the key is turned to Start the engine after warming up the engine and shutting it off. If 12 volts are present, replace the starter. If 12 volts are not present, check the Starter relay.

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This could be almost anything. Does the engine turn but not start when it stops? If so, can you determine if the engine is getting gas?

If you leave it for a while, does it then start and run for a while? If so, a very common problem is the electronic ignition pickup coil. This is the part that replaced points in the distributor. They develop an open when they get hot and when they cool down the connection is made again. This makes the car run until it gets hot and runs again when it cools down.

The good news is the pick up is not very expensive or difficult to change.

Best of luck, Jim

give the engine a minute or two to cool down before turning it off. This cool down period allows oil to circulate and cool down the bearings in the turbo. If you shut off the engine immediately after hard driving, the oil can gum up around the hot bearings and create problems down the road.
Hope this helps

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try changing the coil- sometimes they get hot and break down- once they cool off vehicle will start and run until they get hot again