This is a classic symptom of internal hydraulic leakage inside the transmission. As a transmission gets older, the various seals inside of it that keep fluid confined in specific channels tend to become hardened and no longer pliable enough to seal tightly against the shafts, servos, and other components that they are attached to. Often, in warmer weather they may seal enough to allow the transmission to operate somewhat normally, but when colder weather comes this tends to cause them to shrink more and become less pliable because of the lower temperature. After the engine runs while and the transmission fluid warms up, the internal seals and O-rings expand slightly and become slightly more able to do their job.
You do not mention what vehicle you have, but if this is a Chrysler vehicle, some models (such as mid 90's Intrepids) had other problems such as converter drainback due to faulty seal design from the factory that also contribute to this condition; many Chrysler transmissions have had their seal materials updated to new part numbers to reduce this type of symptom since they were originally manufactured.
The repair needed is a transmission rebuild: removing the transmission from the vehicle, disassembling it completely, and installing all new seals, O-rings, and gaskets. Of course this is pretty expensive because of the amount of labor involved....
As a temporary measure, what often helps is to take the vehicle in to a shop to have the transmission flushed; this involves using a machine to push all of the old worn out transmission fluid out and replace it with new fluid. Transmission fluid contains additives that help to keep seals soft and pliable, so replacing the old worn fluid often helps somewhat. Also, adding one of the various transmission stop-leak or seal conditioner products available at any auto parts store to the fluid sometimes is beneficial. These products work by softening rubber components and causing them to swell up slightly; this improves sealing ability, especially when cold. Keep in mind though that this is not a fix; it only buys some time until transmission overhaul is needed.