Not sure on the 2001 but there should be 3 relays in the engine fuse box that are the same part number.
Fuel Pump(FP) relay and WOT(wide open throttle) relay should be the same, swap them around, that will tell you if FP relay is the issue.
WOT relay only works at WOT, turns off A/C compressor and tells computer to ignore O2 sensors and fuel economy, "driver wants to GO FAST!", lol.
Look at the FP fuse contacts could just be some corrosion in there causing intermittent connection
Computer(PCM) monitors FP power, PCM turns on the FP Relay and then reads if relay closes and FP wire has 12volts on it.
So that could activate the check gauge light, if PCM didn't see the 12v on the FP wire, just guessing that but it would make sense from the wiring diagrams I have seen.
PCM only runs fuel pump for 2 seconds when key is turned on, fuel pump doesn't start again until engine RPMs are above 500, or if key is cycled off then on again, reads like you already know this, but just wanted that to be clear on that part.
The PATS anti-theft system also controls the fuel system, but you should get a flashing "theft" light if that was the issue.
Last edited by RonD; 03-01-2015 at 03:43 PM.