Why does my iPhone 5 die when air temp drops below 50 degrees

My iPhone 5 consistently dies in 40 to 5 degree weather when the battery drops below 50%. It will not turn back on and displays the battery dead, plug into the charger symbol. As soon as I plug it in though it restarts immediately and shows the correct charge. I tried the genius bar and the guy told me there was nothing he could do. There must be something! Any ideas?

I consistently run my battery to the point of dying and then charge it fully. I've also tried resetting to factory new and setting up as a new phone. None of those things has helped.