Yes, batteries do respond badly to cold weather, but their charge returns to being available when the core temperature of the batteries increases. Most electronics has spec sheets that describe the temperature range and for cameras the lower limit is usually 32F. That does not mean the camera is damaged or becomes inoperative, it means that is the lowest temperature which the listed specs are valid. The camera still works below that temperature but they do not guarantee the specs below 32F. If you shoot high shutter speeds in the cold you know exposure is off a bit because the shutter is not quite as fast as it is above 32F.
It cuts both ways, the noise floor lowers also at very low temperatures so shooting long exposures in cold temperatures sees less noise and blooming. AF also is slower in low temperatures because the lubricants in the lens get stiffer(also, another reason battery lose their charge faster besides the chemical process in the battery becoming impeded by cold) Stan St Petersburg Russia