On some models there is a fuel pressure sensor at the end of the fuel rail. This sensor sends a signal to the ECM and the ECM activates a fuel delivery module usually located in the trunk area. The system uses pulse width control to control the fuel pump in the tank. If there is no call to increase/maintain reqd. pressure, the fuel pump does not run or runs very little , so you don't hear it operate some of the times. The module in the trunk just handles the current required to operate the fuel pump.
It might be a good idea to check the connections since when cold things have a tendency to contract and a marginal connection can become an open connection if the temp gets cold enough.
Also you might want to check the crank sensor that is usually located behind the crankshaft pulley at the front of the engine. This could become open when the temp gets cold enough.