When I was growing up friends of mine used to keep their nail polish and batteries in the refrigerator. Why? They thought it would make both things last longer, but as it turns out the opposite is true.
With batteries, cold storage can actually harm their endurance. And with nail polish the cooler temps will make the lacquer thicker and harder to apply. Instead both should be stored a room temperature, and if you can, store nail polish away from direct sunlight as well.