Lithium Energizer AA are the only option for winter use. Cold has no effect on them.
Alkaline batteries are water based and will die in the cold super fast, but can be used in the summer no problem.
Rechargeables are a PITA.
This is an excellent website which discusses this exact topic.
trailcampro.com
They also have reviews on almost all of the game cameras you can imagine. They also discuss in detail and have tested the differences between batteries.
In short, Lithium are the best by far, and the most expensive. They keep a consistant power level to your camera from 100% until they are dead. depending on the camera, you can get close to 1 year on a set of batteries. Best thing I have done to extend life, is use Lithium in the winter on cams, pull them for alkaline or recharageble in the summer.
AS alkaline batteries drain over time they put out less and less power. meaning your flash deminishes as the batteries lose power, also shutter speeds can slow.