Is it an issue to store fluorocarbon in cold temperatures? Like a unheated basement in the winter?

I hope not, mine is in the garage 365 days a year, in northern Illinois so anywhere from around zero to 120 degrees.   LOL

Lol good to hear

https://www.bassresource.com/bass-fishing-forums/topic/207128-shelf-life-of-flurocarbon-and-mono/#:~:text=Fluorocarbon line isn't affected,have years of shelf life.

Fluorocarbon is safe to store outside in the winter, but you'll want to bring any monofilament inside.