2000 Mercury Sable Will not run when cold

Engine Performance problem
2000 Mercury Sable 6 cyl Front Wheel Drive Automatic 135K miles

This car recently sterted having problems starting cold. I noticed that their was no fuel pressure when the car would not start, but later in the day after the weather warms up the car will run with no apparent problems. Next morning when cold again would not start. Changed fuel filter, dropped tank and checked fuel pump filter. The fuel pump runs and can be heard even when the car will not start. Put everything back together, this morning the car started and ran fopr about three minutes and then started running rough and surging until it died. No fuel pressure again. Came back in the afternoon and car started right up and runs great.

I have a new fuel pump but am hesitant to install since I find it hard to believe the fuel pump would act this way. Any thoughts?