Most likely it has nothing at all to do with the transmission. 99% of "hard cold starts " has to due with the battery. Try cleaning the terminals, starter and relay/solenoid connections, and if you still have the problem. It's time for a new battery. Exceptions; if you live in EXTREMELY cold climate's, like Alaska, you may need a electric cord to the engine block to prevent the oil from getting too thick.
You need to check the transmission computer for codes to know possible causes.
Check fluid level, is the vehicle warm or cold when this happens. The torque convertor may be bad. The transmission may need to be serviced (new fluid,filter, etc).
i don't know if this is exactly an answer but I've noticed mine does that on cold mornings (around 32 degrees)
Low fluid level would be the first and simplest thing to check.
There are several possible causes. A few:The engine is cold and making less power so you are pushing the gas peddle further causing the transmission to hold the gear longerThe computer is programed to do this ( but I have no idea why it would be)The transmission has a sticky or stuck valve that requires the higher speed to move.It is really really cold and the oil in the transmission just needs time to loosen up.
Because it is awful...
Automatic choke stuck open
EITHER NEEDS LINKAGE ADJUSTMENT OR TRANSMISSION OR MOTOR MOUNTS COLD HAVE SHIFTED.or wrecked.Or THE DETENT in the trans is sticking
Try to replace the oil filter if the transmission is not engaging when it's cold.
Check engine oil when you have just finished driving. Check Transmission fluid when the car is running and the transmission in in N (manual) or P (automatic).
Through the automatic transmission dipstick tube. Add proper fluid with engine running, transmission in park, until the dipstick reads full cold. Type of fluid is marked on the dipstick.
The automatic choke isn't engaging, the throttle body should almost close down completely when it is cold out.