Will this cause my gun to go rusty?
Coldness on its own does not cause rust, but condensation does. Condensation usually occurs when a mild, damp day follows a period of cold.
To quote an extreme example, during a recent winter the temperature in my outdoor, unheated workshop went down to -14C.
I checked it every day and there were no problems until we had a mild, misty day.
The whole place was suddenly running with condensation, and I mopped about half a pint of water out of the chip tray under my lathe.
Inside a house there is always a danger of dampness on cold days, caused by such things as kitchen and bathroom steam, so it pays to take precautions.
Examine your shotgun regularly and give the metal parts a wipe down with a lightly-oiled cloth.
You can also provide added protection by placing something which will provide an anti-rust atmosphere in your shotgun cabinet.
Check out the Napier range, which is available from most gun shops.