P0125 = Coolant Temperature Signal Rationality
In the absence of other operational or driveability issues...
First, check the wiring harness and connector on the Engine Coolant Temp (ECT) sensor. It's located right near the thermostat housing.
Next, test (or just outright replace) the ECT sensor.
Then if that doesn't correct the problem, go ahead and replace the thermostat.
Lastly, there's a very slight chance the Intake Air Temp (IAT) sensor is giving false readings tricking the Powertrain Control Module (PCM) into thinking the ECT and IAT sensor outputs don't properly match under certain conditions.
Best regards,
John L.