My early 90’s Corolla has that symptom too. I know what is causing it, the idle air control is misbehaving. That part is difficult to source so I just live with it. Living in San Jose, except mid-winter it is seldom cold enough to cause any symptoms that warming up the engine in the driveway a few minutes doesn’t solve. But as part of diagnosing the problem I investigated the throttle position sensor, the computer’s coolant temp sensor, and I removed and cleaned the backside throttle body. So based on that experience, here’s what I’m thinking is happening in your case …
It’s possible to test the coolant temp sensor (the one used by the computer) if you can find a chart for its resistance vs temperature. Then all you have to do is measure the resistance and compare it against that chart… If you don’t know how to measure resistance, probably best to just replace it. Could work.
My guess however is something amiss with the throttle valve. So if the coolant sensor doesn’t pan out check the throttle position sensor, and after that, probably time to remove the throttle body and clean out the back side. It tends to get gunked up with pcv and egr gunk, and can make the throttle a little sticky, especially when it is cold.
If there’s anything tune-up related that’s due, I’d do that before anything above. For a 2003 – I’m assuming the engine is configured with COPs – so a tune-up would probably mean just new spark plugs, new engine air filter, check the idle rpm and base timing.