Is there any effect on fluorocarbon line when the weather is cold? I'm talking about temp. in the 30's to 40's.

I definitely think so, being a more dense line.  I purchased some Inviz-X just for the first month or so of fishing, when the water temps are in the 40's.  The Inviz-X stretches quite a bit more than the Vanish fluorocarbon I normally use, and should work well.  Breakage may not even be an issue; I'm just taking precautions.

Mostly it's about memory.

Many lines, especially pure fluorocarbons, copolymers and thick monofilaments, become slinkys in cold weather.

what should a person do in these conditions?

pitch

Lighter lines,

trilene made a cold weather line, not sure if they still do

use braid, temp has no efect on it

Cold weather does affect braid. I was fishing in the low thirtes and my braid actually froze

making it stiff like wire. I had to switch to mono that day.

Due to me growing up wisconsin iam around ice fisherman. I have never heard any thing bad about braid in ice. Some guys use it and swear by it. I have never had braid freeze on me but iam useing power pro. I have also heard a lot of good things with fluro and ice fishing.