Hi Anne: The danger to tools exposed to cold depends on two things: what the tool is made of and how much moisture is in the air.
As the temperature drops, the ability of the air to hold moisture also declines. Moisture will then condense on the coolest objects first, which around the house usually means glass and metal objects. The best rememdy is to keep the area heated or extract the moisture using a dehumidifier (running your central heat fan if you have an central air conditioner will act as a dehumidifier).
Hardened steel is the most sensitive (most prone to rust and corrode) to cool/cold weather. Wood, cloth, non-ferrous (aluminum, brass, copper, etc) metals do not seem to be too adversely affected by being kept in cool weather, although I have in the past treated my wooden handles with linseed oil each fall with what appears to be good results. For hard steel (wood cutting blades, router bits, wood chisles, tables saw tops, etc, etc), for the past several years, I have wrapped or covered these with old quilts, blankets or other soft fabric cloths, as all my tools are kept in my 12' X 16' unheated shop that is unattached to our house. I do not coat them with oil, as most oils are hydroscopic (collect moisture) and it tends to be messy. By wrapping them, it seems to protect them from the condensing moisture in the air.