I have an 01S with 150K. Last year, the CPS failed. Symptoms were: 1) start car, 2) run to get up to operating temperature, 3) purposely shut off engine, 4) try to restart, no joy. If I let the car coll down for 20 minutes, the engine would restart. Replaced the CPS and never saw the problem again.
So the cold / hot phenomena is real.
Try the two tests and I'll bet you'll find a bad CPS.
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