Power to the fuel pump

bad fuel pump or fuel pressure regulator.

do you have a spark at the plugs ? if so ..main cause is fuel pump connector plug under car by fuel tank . test this bye getting a Friend to turn ignition on but not start whilst you listen under car for fuel pump buzz. if you hear a buzz forget solution 1 if no sound try solution 1. move the plug (4wires in front of fuel tank ) about then try again . if you have a tester and power is passing this point (pump side) then it fuel pump faulty.. solution 2 crank sensor which is on the lower half of the engine block .. hope this helps

Generally speaking, its supposed to do that.
The fuel pump(s) is powered by a relay that turns on for 3 seconds when you turn the key on. If the engine does not start and produce oil pressure, thats all the power the pump gets. If you were in a wreck the pump would shut off when the engine dies even with the key on to prevent a fire.
You would need to jumper the relay to make the pump run beyond the 3 second time.

Did you reset the inertia switch that shuts pump down in case of collision?

Voltage drop test the electrical circuit , do know how to do automotive electrical testing . Lose connection , resistance in the circuit , lose ground Mastering Voltage Drop Testing with Pete Meier and Jerry Truglia Testing Electric Fuel Pump Problems with Voltage Drop Test

I'd have to test when problem is ongoing. When you first turn on key, no crank, you should hear the pump run for a few seconds then stop. For initial prime, the computer doesn't need rpm signal. Engine cranking, the computer needs to see rpm signal or it won't ground the fuel pump relay. W/o rpm signal, no spark at the plugs and no fuel injector pulse. I know your talking about fuel, but, when problem is ongoing, check spark and fuel injector pulse, engine cranking.
Intermittent problems are a pain, have to test when problem is ongoing.

Not guessing as to what the problem is , testing for voltage drop problems with a volt meter . posted a video on youtube yesterday . If your still looking for guesses forget it , no one on here could say for sure without testing the circuit an hooking up a scan tool to see if PCM is energizing the relay . Do yourself a favor an take it to a qualified repair shop . Fuel Pump Electrical Circuits Description and Operation If you can't learn how to test from this guy , better take to a repair shop .

replace the oil sending pressure sensor on the engine which provides the priming of the fuel pump uopn first starting the engine

you should change that relay

Was it the fuel pump? If not I would consider the charging system ie) battery, alternator or both.