Like all batteries, they do not.
And when the battery booster itself is cold, even if it is fully charged, its power is greatly reduced.
But storing them indoor is kinda pointless, cause these portable boosters are supposed to save your day when your car's battery dies, so they are useless if you are out and the battery dies while the booster is home.
And unfortunately, usually your car's battery dies on the coldest days of the year, so a lot of times these battery boosters are just 'better than nothing' last resort but don't hold out too much hope for them. If you have a 2nd car around, that is always a better option with jumper cables. Always have a set of jumper cables in the trunk if your car battery is old.
What you need to do for these battery boosters in winter is keep them charged constantly. I charge them monthly to keep the charge up. And if the car battery breaks down where you have access to indoor (home, office, etc.), bring the battery booster indoor, and let it warm up to room temperature before you try to boost the dead battery.