+1 on letting the saw warm up, but not just in the cold.
Give it a Min. or two for everything to warm up to a uniform Temp. before givin' her hell in any weather.
If it's below freezing, you should be fine with most GOOD Summer weight Bar oils untill it gets down around 20 or so from what I have seen.
Just watch the 'House brand" stuff. If it's as stiff as 80-90 gear lube when it comes out of the jug, forget it, or thin it out with diesel or motor oil.
If ya don't cut much in the cold, a quart of winter weight on hand, to thin down the summer weight stock is cheap enough. Just in case ya have a buddy call ya in January with a tree on his house or somethin'.
Stay safe!
Dingeryote