When it’s cold outside my car will start up, but not stay running. The only way I can get it to stay running is to keep my food on the gas for up to five minutes. Once the vehicle is warmed up, it stays running. If I start it every hour or so, it will run fine. It’s just when it’s been sitting out in around zero degree weather overnight for example.
Sounds like something wrong with the cold start stuff. On cold starts extra gas is supposed to be injected. For this to work the car must know the coolant is cold, so the coolant temp sensor is one place to look. If the extra gas is injected by a cold start injector, that’s another place. On newer cars there’s no cold start injector. Instead, the computer tells the injectors to ante-up some more gas on cold starts. So the computer (ECM) or the injectors are another possibility.
I’d start with the coolant temp sensor or work your way from there.
Also, if your car is behind on scheduled maintenance, now’s a good time to bring it up to date.
You are probably looking at some failing component that can only be diagnosed in failure mode. So to fix it properly it should be left overnight at the car repair person of choice to be able to witness and diagnose the problem. You may wish to investigate a block heater.
If you have the 4.0L SOHC engine, I can say with 90% certainty the problem can be the intake manifold seals. The seals from the factory were faulty, and lead to leaks and hard starting when the engine was cold and the temps were low. You should also be getting a check engine light and a DTC code of P0171 and/or P0174 for system too lean. Most of the seals lasted past the factory warranty, but most crapped out after 9 to 10 years. Ford has a TSB for it, but I don’t remember the ID number.
@BustedKnuckles is right that anything allowing extra air to leak into the intake airstream can cause this symptom. Excess unmetered air leans the fuel/air mixture. I’d expect some warm-engine drivability symptoms too if this is the case. But the symptoms are worse when cold b/c the engine needs a richer mixture when cold.