Battery problem

It is unlikely the cold damaged your phone unless it was below 0F and being used.

If nothing with regards to your usage has changed and you have not noticed a change in the ambient signal (there's always a chance your nearest tower is down for maintenance), then the first step to troubleshoot is to attempt a restore via iTunes. That should help isolate a software vs. hardware issue.

 

I've taken my phone out into the cold before (using trailhead for a winter weather hike) with no issues or ramifications to the battery life. Probably unreleated to cold IMO.

Do as the above poster said. Try to rule out software vs. hardware problem. If software, come back and we can help you to further narrow the issue. If hardware to the apple store you go.