My car doesn't start on hot days

Change the thermostat,not opening all the way.

Need a fan to cool it down.

You will need to explain all that at a repair shop,
so they can duplicate the issue & then do some
testing, to find & resolve the problem
No one can tell you over the internet on a guess
Always need the car to find any problems

some times when the water pump is changed you gotta fill up the radiator with coolant, run the engine until it gets hot, shut off the car and wait until the car cools down to put more coolant because some air may stay trapped inside and by following this steps you get rid of the air trapped inside.
try this:
1) when the car is cool take the cap of the radiator off and see if it's full. if it isn't, with the car still off, put some coolant until it gets full.

2) once the radiator is full, start your car and wait to see if the coolant goes down the radiator. if it does, keep putting coolant until it doesn't take any more.

Hope to be useful.

check the starter relay contacts and the electrical portion of the ignition switch. the heat can expand a solder point just enough to create an open circuit.

is the car running hot? some cooling fans never come on going down the road because the air comjng through the front of the vehicle keeps the car cool enough that it doesn't need to come on. but if it is overheating then you may want to check the cooling fan relay.

It sounds like your AC compressor bearings may be overheating causing excessive drag/loading on the engine. No easy way to check this unless you can take the serpentine belt off when the car won't start and see if it is really hard to turn the compressor shaft.

Has Neon do same, I ran fuel at low level, on hot day, ran engine until it started to act up & threw a fuel pressure gauge on it. Was at approx 5 psi. Cold next morning, 48-50 psi. Always hot day about 30-40 miles into work in morning, o.k. Sit in hot sun all day & would happen 3-4 times on way home. Sit and let cool, 20 minutes or so & go again for a while. Cool day, no problem. Change pump I would.

Because this communicates on RF, I would suspect that there is loss of signal when overheated. I also am in AZ and see CAN buss and RF go down in heat. This time of year you should be able to leave it @ shop to recreate problem and then check for buss. After alll, it was 107 here ysterday.

you have a short circuit somewhere in your dash cluster, when it heats up the wires probably soften and droop down contacting another wire and closing one or more lamp circuits. If you feel like taking the dash apart, get a wire location diagram and find the short, or take it to an automotive electrical shop.