Cold, in and of itself, doesn't appear to do any harm.

Rapid transitions from cold to warm will often cause the finish of a guitar to "craze" - i. e. get a whole lot of little tiny cracks, just in the varnish, not affecting the wood.

This doesn't really harm the guitar at all, and some very knowledgeable people claim it can improve the sound of a guitar, by letting the wood vibrate more freely. But a lot of people are horrified when it happens, because the instrument looks beat-up.

To avoid this happening, have your instrument in a good hard case, preferably with an insulated case cover, when you take it out into the cold; and don't bring it into a warm room and immediately take it out to play it. Let it warm up to room temperature inside the case.

Peter.