I have seen several posts on this issue but haven't seen anything that exactly matches. I have a 02 2.3L automatic that has had this issue for about a year and a half. After short drives if I stop somewhere, sometimes it won't restart until it sits for a little while. All of the dash lights are there, it cranks but doesn't fire. It has shut off while driving, the green key light will flash and other lights come on, and after sitting a few minutes it will restart. Most recently, after sitting outside on a pretty hot day it wouldn't start, again turning over but not starting. We did a crude check for spark from the wires on two cylinders and it didn't seem to get spark. The next day started right up and drove home with no issues. I have replaced the valve cover gasket and spark plug gaskets, spark plugs, wires, and distributor cap and rotor. Also checked fuses and the relay that's in the car under the steering wheel. Did the check where you unplug it, turn the key on. Plug it back in and everything seemed normal. This car has over 200k miles, but runs fine beside this issue. Sorry for the length but I wanted to put in as much detail as possible.

Thanks in advance
***UPDATE***
Took it to the Honda dealer yesterday. Tested immobilizer and they said no issues. They did run some other test and got the car to stall ay 3500rpm. I do have a crack in my air intake hose, and they recommended replacing it and cleaning the throttle body. They could not replicate the crank but no start issue and have no clue what it could be.

 

Thermal failures can be a right pain in the hole to diagnose.
Verify that the battery connections are clean and tight, and that the battery is fully charged and does not have excessive voltage drop during cranking.
Check for any stored codes, just because the CEL is not on does not mean there is not a problem(code) stored in the PCM.

When the car does not start and the engine is cranking does the tach move?
If not then the ICM inside the distributor may be dying, it sends the signal to the coil to fire and signals the tach.
For now on a hot day to mitigate the thermal failure, when the car won't start, pop the hood and let the engine bay air out/cool down a bit then attempt to start.

 

If, when the car dies, the green light is flashing that means the immobilizer isn't reading the key. You may need to have a new immobilizer installed, but first try another key. If you don't have another key, buy another one and have it flashed to the ecu. Then try that one. Perhaps it's just your existing key that's not able to be read sometimes.

 

If it's the key I don't think it would correlate with this only happening when it is hot out, or after restarting the car after a short drive. I'll give it a shot though.

 

So today, without being particularly hot out, I felt the car hesitate a couple times, and the green key light came up solid. Then as i was turning just momentsb later the car shut off. Several lights were on, and i think the key light was on again. The key in the ignition was not the one i usually use, and when i went to restart it cranked but no start. Switched back to my usual key a second after, and it started. Coincidence? I'm not sure. Do you guys think the immobilizer or ignition might be the culprit after that?

 

The keys would be unlikely to all go bad at the same time. More likely the ignition switch itself or the main relay. Bushougoma did a good YouTube video on how to diagnose this