To keep your device running smoothly when the mercury dips, get a case if you don't already have one. Beyond protecting your precious screen, it also insulates the machine, helping it retain any heat it generates.
If you know you're going to be out in the brutal cold for a while and can afford to turn your phone off, by all means, power it down. Even though your battery hates being exposed to sub-freezing temps while it's running, it's heartier when turned off, capable of safely weathering up to -4 degrees without going bonkers.
It's also a good move to keep it closer to your body in an interior jacket pocket -- and maybe control the urge to scroll through your Instagram feed until you're indoors. Your fingers will also thank you.