2003 Jetta 1.8T problem...need help

I was just wondering if anyone knows what this is or how to fix it. Here's the story. I just got my 2003 Jetta 1.8T this fall and it is great. This winter on the really cold mornings I have trouble starting it. I put the key in and turn it, but all that happens is the radio goes on and heat/defrosters go on. When I turn the key there is no turnover or clicking what so ever. I take the key out and try it again until usually about 3-5 tries later it finally starts up. I replaced the battery a few weeks ago and everything worked great, until it got colder again. So this morning the same thing happened. It took 3 tries of taking the key out, putting it back in the ignition, and turning it again with absolutely no clicking or turnover. Finally on the 4th try it started. It only seems to do this on the coldest days, and after it starts once for the day it starts up perfectly every time after that the same day, no matter how long between starts. I can't figure this out. Please help!