It is very hard to start when the engine is hot after a short stop. Sometimes I could keep it running pushing, pumping the gas pedal very fast, after 5 minutes struggling with the engine sputtering knocking running very rough and wants to stall it will start to run smooth. But if it stalls I cannot start it for 40 minutes but it is running fine after that. I do not hear the fuel pump ever running after I turn the ignition on. Thanks for your suggestions. The engine has 130K miles.
There might be fuel control problems (pressure, and spray). The repair manual will detail how to check these (fuel pressure regulator, fuel injectors, sensors).
You can improve air flow (which mixes with the fuel flow) by using a Throttle Body Cleaner on the MAP Sensor, idle air control valve and its air passages, the throttle body bore and throttle plate, and a clean air filter.
You did check fuel pressure, right? How many PSI was it?
And if it was OK, I suspect coil packs (aka Distributorless Ignition System module), But don’t replace it unless you have erratic spark as determined with an inductive timing light.