This is the EXACT problem I had when trying to make a "fridge temperature puck" - batteries work less well in very cold situations, probably due to low temperature decreasing molecular reactions. I was planning on having an Arduino collect stats and then display them as a webpage over Wifi.

There are basically 2 solutions, and I feel both are inelegant:

1. Use part of the battery voltage in a heater that warms the battery enough for it to continue reacting and producing voltage.

2. Locate the battery outside of the fridge somehow, with a long wire to the rest of the device.