How 'bout a worn out fuel pump that has 'morning sickness'? If the pump is on its last legs(how many miles on the pump?), it may have higher resistance across the brushes, or more friction in the bearings that loosens up as the pump warms, or makes better contact.
Was the volume checked when the pump was cold?
The things that affect cold starting are multiple, but the major one is fuel:air mix. The fuel has to atomize and mix with the air to be combustible. If it is not mixed, due to bad spritzers, or low pressure, or the vapor pressure of the fuel is set to 'summer', it will be hard to start. If you cycle the key to do multiple 'prime' cycles with no difference, and the fuel pressure is maintained, or is at 40+ after the FIRST cycle, then I'd agree the fuel delivery pressure and availability to the injectors should be sufficient. How old is the gas? How cold is cold? Have you tried other brands of fuel? Are you using tractor gas? I don't even know if there is still such a thing...
tom