There are a couple of possible issues here:
1. When the battery is cold (freezing or lower), it's more likely to throw a dendrite. That will shut the phone off. (See
Lithium Battery Dendrite links.)
2. You should never allow the battery to drop lower than 40% - start looking for an outlet when it gets down to about 50%. (Or carry a power pack or, if the battery is removable, a spare, fully-charged battery.) Sorry, people - lithium batteries are
not deep-discharge batteries. If you want to drain the battery to almost 0 every time, without destroying it, you'll have to carry a lead-acid battery with you. But one to power the average phone will weigh about 5 pounds.
3. Because of #2, and the fact that the phone's shutoff point can be set, set it to about 38%. If you absolutely must drop it down below 40%, don't let it stay there too long.