I HAVE THE SAME PROBLEM BUT NO ONE SEEMS TO KNOW WHAT PROBLEM IS
I also have the same problem with a 2001 GS430 for about 5 months. It also seems to take longer to start working in colder weather. I also noticed I can sometimes get it to start up right away if I restart the car about 4 or 5 times. No luck in diagnosing the cause of the problem. Whenever I take it to the dealer the problem seems to go away by the time I get there and then they start the car and it works normally that day. My wife claims it is possessed, but I suspect there is a relay that gets energized by the ignition switch that just hangs up in cold weather. If anyone has a fix I would really like to hear about it!