A cooling apparatus for an engine is provided with a thermostat operating in such a manner as to control a supply of a coolant to a radiator. An electronic control apparatus estimates a reference temperature corresponding to a temperature of the coolant on the basis of at least a vehicle speed, and diagnoses an operating state of the thermostat on the basis of a comparison between a detected value of the coolant temperature and the reference temperature. The electronic control apparatus inhibits the diagnosis of the operating state of the thermostat in the case that any one of the following conditions i) to iii) is satisfied. i) a rate of a descending slope traveling time of a vehicle with respect to a time until the diagnosis condition is established from the start of the internal combustion engine is greater than or equal to a determination value; ii) a rate of the descending slope traveling time with respect to a general traveling time is greater than or equal to a determination value; and iii) a rate of a time of a specific state (a state in which a speed of the vehicle is greater than or equal to a reference vehicle speed and a load of the internal combustion engine is less than a reference load) with respect to the time until the diagnosis condition is established from the start of the internal combustion engine is greater than or equal to a determination value.