Does it make sense that my car would shift correctly only when cold?

This seems to be the case, I get longer held and smoother shifts when I first take off, then the hotter the engine gets the faster it goes through the gears and the rougher they hit. Any ideas?

Would a thermostat modification, causing the car to take longer to warm up, fix my problem or does it sound like the transmission oil cooler is toast?

Sorry if I asked too many questions in one thread, but I'm dieing to know!

Thank you for ALL replys and advice!