Starter?

Severely cold temp can cause the battery to fail, but not the starter. However, I've seen some really strange things happen, for example: moister will get into engine compartment and wires/connectors. This moisture will freeze overnight and expand. This will cause all kinds of issues that are hard to track down.

Also, when ignition locks and starters begin to fail, they do it intermittently. Eventually they fail all the way. First check the tightness of all your bat cables and for corrosion. Not just at the battery but the frame, engine block and starter.

Finally, if you get any check engine lights, have the computer scanned.