According to the Service Manual for a 2000 5VZ-FE (same engine, I think), the P0125 code actually has nothing to do with the coolant temp, even though that's the name of the code!
Possible causes for P0125 include a problem with air flow sensor Bank 1, Sensor 1 (that's a heated oxygen sensor, I believe), fuel pressure, or an injector.
P1135 is a malfunction in the heater for the air flow sensor Bank 1, Sensor 1, indicating that either the sensor's heater is bad or the circuit supplying power to the heater has a problem.
Together, it sounds like the sensor.