This issue might take some step-by-step diagnosis to find the source of the issue. Since it sounds like a repeatable issue, it should be able to come up with an educated root cause (instead of just throwing parts at the issue). I think Peter's suggestion is a good place to start (finding any fault codes in the ECU). After that, I would suggest testing the fuel pressure when the engine is cold, hot, and in between and checking it against the specifications. Then, get the engine warmed up with the fuel pressure gauge attached, then shut off the engine and see what happens. If it drops fast (within a minute) then the fuel injectors might be leaking. You can rent a fuel pressure gauge at most auto parts stores. Fuel injectors can be cleaned professionally (for around $20 each), but new ones are only around $40-$50 each, new.
It could also be a non-functioning check valve in the fuel pump, which is supposed to hold pressure in the fuel lines. You can try to pressurize the system (by turning the key to the 'run' position) before trying to start the car. Do this a few times before cranking, just to be sure. If the problem goes away, suspect the check valve. I suspect leaking fuel injectors, though.