I was wondering, why is it that when I run in cold weather I get a sore throat. The pain is always located around the same area where a tracheotomy would be done. I thought it might be because sometimes while I run I breath through my mouth too much, but then remembered that it only happens when it's cold, and not during normal or hot temperatures, eventhough the way I breathe while I run never changes.

Cold air rapidly breathed in is not sufficiently warmed before it hits the back of your throat, causing pain and discomfort.
Try wearing a thin scarf, balclava or neck warmer over your mouth on cold days. This will pre-warm the air before it enters your throat. You will figure a happy medium between increased warmth and decreased intake volume pretty quickly.
Generally, I opt for the scarf or balclava when the temp gets down to single digits Farenheit. I've run down to -40F (-60 with wind chill) while wearing a couple layers of polypro long underwear, a wind shell, hat down to eyebrows, scarf up to bottom of the eyes, thick socks and wind stopper gloves. I couldn't do the run without the scarf wrapped twice over my nose and mouth.
Now, the real challenge is if you CrossFit inside for the cals/lifts and outside for the run in the same workout. I haven't worked that one out so I either get sick or sub something else for the outside run so I can stay inside in my relatively warm garage.

Great, thanks for the advice. Being originally from Puerto Rico and currently living in Florida I never have, and can't even imagine running in -40 degree weather. I'm sure I'll be doing that someday, maybe not in -40 degree weather, but who knows. Again thanks for the advice.