Engine will crank but not start when temperature is below freezing

There may be a faulty coolant temp sensor that is not telling the ECU how cold it is and it does not know how much to lean out the mixture to start the car. There is a part called an Idle Air Control Valve that nadles this and keeps the idle steady in all conditions. It is either getting the wrong signal, has a loose wire and not getting a signal or it is not moving the solenoid that it uses to change the size of the port that lets it work like a, "choke" by leaning or richening the mixture.
There could also be an air pocket in the cooling sytem. Try to check it by starting the car when it is cool and turning the heater all the way up. Wait for it to reach operating temperature and top it off. Replace the cap and repeat once.