Frost on the inside of the barrel does leade to rust in the inside of the barrel. I have bought a lot of used guns for the actions only and it is amazing how many of them have surface rust in the first inch of the barrel(walking with the muzzle up in the rain).
When the hunter comes home from his hunt, he brings his gun in the home where it is warm, rust develops very fast especially if he puts it near the fire place.
