The fuel pump will ALWAYS kick on when keyed on. No matter the temperature to ensure there is pressure at the rail. The engine needs ~30psi (correct me if wrong) for the engine to properly fire. My 03 would still run (like sh!t mind you) when the pump died as it was siphoning fuel on its own. Once you get it running is it running as if there's no problem or is it missing and bucking?
Put a pressure gauge on the schrader valve on the rail and see what kind of numbers you're getting.