It may be more complex, but I'd replace the battery in the key fob as a start off.
When the batteries get low the car doesn't get the signal from the key to tell it that the key is the right one.
Also, when the key gets physically cold, the battery doesn't produce power as well - which you'd only notice in the winter.
If you warm the key up in your hand and it works, it's a cert to be a low batt.