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The thermostat monitor is a function of the PCM and is designed to verify correct thermostat operation.
The monitor executes once per drive cycle and has a monitor run duration of 300-800 seconds. If a
malfunction occurs, DTC P0125 or P0128 sets, and the MIL illuminates.
My experience is that a faulty thermostat would not give you a code but substantial overheating would. The "thermostat monitor" function you reference is news to me, and you'd have to research the parameters that would set the code. I suspect that in some situations it may not indicate the defect.
A new thermostat in less than $10 and changing it only takes a few minutes. The symptoms you describe would point to a faulty or sticking thermostat. Less likely would be a faulty fan clutch, but that would usually give you a code.