If you have a single wire sensor, the ECU will wait a certain amount of time before checking the sensor to make sure it is functioning properly. If its too cold, it won't work right, and the ECU will turn on the engine light and go into a slightly rich operating mode. It is possible that if the O2 sensor is on the cusp of not working that the engine would run richer then normal. Most likely the ECU will just thinks its broke though.
If there are problems, you may be able to substitute a 3 wire O2 sensor with a heater. I believe they are self regulating and only require 12v to the heater circuit when the ignition turns on.