In theory, it shouldn't start or run when it's cold outside. And IF it does, compressor damage can happen because it doen't have oil in it designed for low temps. In addition, refrigerators lack a compressor heater to keep the oil preheated so it flows. Those with heat pumps will see this compressor heater band if they look at it. Same for HVAC heat pumps. They have a heater so the compressor will start when it's cold outside. When it doesn't, it's time for the emergency electrical heat strips. :-/
Personally, I would unplug it, dry it out, install a box of backing soda in it and shut the door. If it's a small fridge, I'd put it in the garage with the door cracked open. Again, also unplugged. Refrigerators are not designed to run in 30degree weather even inside your cold garage.