Right off it sounds like a component gets too warm and fails, but works cold. Is the engine getting fuel when warm? Does it have spark when warm? Knowing the answers to those questions as well as make, model, year and engine size would help to answer your question.
I had a similar problem with a 1987 Chevy Celebrity. I replaced the ignition module and the crankshaft sensor and it runs fine now.
it sounds like your choke is not operating properly.
Moisture is getting into the distributor cap. It evaporates when it gets warm out. Replace it.
check cold start sensor or cold start valve is the same dif. name, also check for the coil pack could be in short good luck
You need to warm the car up with hot water.
It starts cold
This problems could be the injectors. However it may also be the coil if the coli is going bad it will does this constantly.
check coolant temp sensor
I have a Toyota with the 22r engine and it will start when its cold but it when it gets worm it wont start with out help
around 30 or 40
the starter may be worn on your car if it starts up easily when its cold but slowly turns over when it is at the operating temperature. Replace the starter to your engine.
try checking your battery usually it is the lack of cold cranking amps
What probly is going on is that your ignition module is failing. You can take this part off and have it tested, but the test will more than likely be false because, like you said, it only fails when warm. Start with replacing it with a junkyard part. This way, if it doesnt solve the problem, your not out the 100 bucks that the parts stores want for it.