Running drain's through an unheated area is not normally a problem.
Drains installed properly either have running water in them or, drain dry after use. Hence they will not freeze.
If the pipes are clogged, have a sag that holds water, or, there is a low volume of water leaking down the drain, I.E. dripping faucet, or leaking flapper, there may be a possibility of freezing as the volume of water will be insufficient to stay warm before it freezes.