My friend's been having issues with her 5 for the same reasons. It's been exceptionally cold in the NYC area recently, but the issue also arised once around 45 F during a Fall hike (60% battery -> instant shutdown). It was notable because my 5 side by side to hers was just fine. It never happened to her again, but now that winter's come around, she doesn't live in the safest area, and it's done it multiple times on her way home during the night. She's not very comfortable with tech, so she asked me to go with her to her Genius appointment. I was pretty confident that they would help her out since I've always had good experiences, especially if I had AppleCare (which she has).
Nope. The guy pretty much denied ever having heard of this problem. And I could understand if he gave a legitimate reason that maybe the ambient temperature was below its operating range, but instead he tried to pull the wool over our eyes with techno babble with some of the craziest explanations as to why it would shut down, along the lines of "She had too many apps open, and her screen brightness was too high, and that can drain your battery from 60% to 0 instantly." "Even though if we warm it up, and it powers right back at 60% battery?" "Yes." "So, doesn't that mean the power wasn't consumed at all? Because it's still there upon restart, and it's usable so long as the temperature is maintained." "Well, when you use too many apps, and the screen brightness is high..." repeat ad naseum. It might have been less insulting if I didn't have a computer engineering background, but I kept that tidbit to myself. Needless to say, I didn't want to outright say "You're full of ****" but after trying to have him clarify multiple times how apps and screen brightness could cause a 60% instant power drain (as he explained it), he wouldn't budge past his rationale and demanded $80 in order to do anything.
Tl;dr: The 5 series and beyond have varying battery vendors and some don't perform as well as others in the cold. Luck of the draw I guess.