I would suspect the battery. Here's the list of things of things that will prevent the remote start and the one that jumps out to me as something that could happen in cold weather is battey charge level. Next time it's cold and doesn't work, I'd go out and and check the battery voltage.
All of the following conditions must be met
before the engine will remote start:
Gear selector in PARK
Doors closed
Hood closed
Hazard switch off
Brake switch inactive (brake pedal not
pressed)
Battery at an acceptable charge level
PANIC button not pushed
Fuel meets minimum requirement
System not disabled from previous remote
start event
Vehicle security alarm not active
Malfunction Indicator Light (MIL) is not illuminated
All of the following conditions must be met
before the engine will remote start:
Gear selector in PARK
Doors closed
Hood closed
Hazard switch off
Brake switch inactive (brake pedal not
pressed)
Battery at an acceptable charge level
PANIC button not pushed
Fuel meets minimum requirement
System not disabled from previous remote
start event
Vehicle security alarm not active
Malfunction Indicator Light (MIL) is not illuminated