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My 2006 has the Electronic Stability Control, and on cold starts the idiot light for this goes on and won't go off. If I turn the engine off and restart it after a few minutes, the light does not come on. Any ideas?
Freaky...this happened in my wife's 2005 New Beetle, just as you described it. Eventually it came on and stayed on all the time. Turned out to be a bad wiring harness. Apparently it was damaged during installation at the factory. I guess that's one possibility.
Of course, there might actually be a problem with the ESC system. I don't know anything about ESC, but I would guess that a problem would set a trouble code. This might be one that AutoZone can't pull for you, but I'd start there. If no code, you should probably take it to the dealer (or a good independent shop) and have them test the system.
Crazy! It only does this when it is cold outside. After driving it for a while, if you shut the engine off and restart it, the light is off! I guess that is why people refer to these lights as idiot lights.......................