It sounds like the fuel pump is being *told* to stop running so we need to find out why. I've attached the EWD for the ECU (TMA - too many acronyms).
I wish I had the Adobe Acrobat so that I can break up the pages. Anywho, on pdf page #8, the "circuit opening relay (COR)" is responsible for the activating the Fuel pump relay (FPR). The COR is getting power from two different sources: Follow the B-L wire all the way to page #3 and you'll notice it is get fed by a 7.5a IGN fuse and the IGN switch. The other end of COR is getting fed by the R-W wire which is coming from ECU pin # 14, labeled "FC" terminal, on page 7.
copy/paste from page 2 of EWD:
FUEL PRESSURE CONTROL SYSTEM
The fule pressure up system causes the VSV (fuel pressure control) to come on for high temp, starts and immediately after starting in order to increase the fuel pressure, improve start ability at high temps and provide stable idling. The engine control module evaluates the signals from each sensor (input signal from (1), (2), (5), (11)), outputs current to terminal FPU and controls the VSV.
Fuel pump control system - the engine control module operation outputs to the terminal FPR and controls the fuel pump relay (FPR) and thus controls the fuel pump drive speed in response to conditions.