I would think a larger concern on the longevity of your shotgun would be the condensation that occurs after brining in your gun from the cold. A cold gun brought into a warm environment will attract condensation throughout its innards as well as its exterior. It is that moisture that freezes the next time the gun is taken outside - usually not the oil. When hunting in the cold, try not to bring the gun into a warm environment until you are done.
Pretty much any light weight synthetic oil will be OK. That is as long as you do not give your weapon a bath in it. I do not know the freezing point of various commercial lubricants, but according to WikiAnswers.com, regular 5W30 motor oil will not freeze until -30F - and that is actual temp not a wind chill. Of course, if you really want to be in the cold graphite will work too.