The safest thing to do in cold weather is to let your flute warm up slowly.  Just open the case up and let it warm up to around room temperature before manipulating the keys.  When your flute mechanism gets cold, the silver or nickel key work will shrink more, and more quickly than the steel rods.  The shrinking compresses the oil within the key mechanism.  As long as the mechanism warms up slowly, the oil will uncompress with it; however, if you manipulate the keys when they are cold, and the oil is under compression, the added kinetic energy from the friction caused by the key movement will overcome the static compression and the oil will expand [uncompress] and begin to flow.  Expelling it from the key mechanism through the key joints.  It is possible to lose up to half the oil from the key mechanism in one incident, cutting the COA interval in half.

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I recommend using an insulated case cover for the thermal insulation to slow down the cooling process.  My picks are Altieri and Olathe, both made in the USA.