Hard to start in the morning

Ever since it's been colder outside my car has been harder to start. Only on the coldest days will it have trouble, when I try to start it, it will sit there and shake while i have the key turned...after about a minute of giving it gas while turning the key it finally turns over. I've already replaced the fuel filter, fuel pump, and the coolant sensor(it was reading -32 degrees and we thought the sensor was the problem before and it did work for a while and now I have the same problem) if anyone has any ideas to what the problem might be I'd appreciate it, thanks

The plugs were replaced about 5 thousand miles ago, the wires could use a change, and I don't even know how old the cap and rotor is, and it still cranks at normal speed....what i don't understand is when in the morning when i turn the switch i don't hear the fuel pump kick on, but anytime later in the day i hear it kick on and it starts up just fine