The ECT ( engine coolant temperature sensor /Computer switch ) tells the ECM ( engine control module ) the temperature of the engine based on that the engine module adjust the idle ,if the engine not showing being cold the ecm is not idle the engine high enough to keep it running,especially if you giving it gas
you basically doing the ecm job and keep it running and when the engine reach operating temperature the ecm goes into closed loop status, meaning switching to the the info on the air to fuel ratio from the Heated oxygen sensor not from the ect sensor after warmed up and oxygen sensors are heated.
But The vehicle needs to be hooked up to a diagnostic scanner to Retrieve any fault codes in the pending file , even if the light not on and the ect can be read on the scanner to what temp is reporting to confirm
From what you describing to me , I think that is what is happening
The idle air control valve sometimes are an issue but Usually it will be doing it regardless of engine cold or hot ,but that will be another area to look at .
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