Car wont turn over? Could it be the cold?

Ok so for a few weeks my car wouldn't start. It would turn over but not click on and the battery light was on. So we jumped it and took it to the auto store where they tested the alternator and battery. Alternator was fine but battery was junk. Bought new battery and it was fine for about 24 hours. Now its doing the same thing again. Whats wrong? Could it have anything to do with the fact that is -2 out right now? Could it be the fuel pump?