Malfunction of an engine coolant thermostat is detected from the engine side coolant temperature by the subsequent behavior of the coolant temperature. When an open-malfunction occurs, the coolant temperature becomes considerably different from that normally expected for times at which the thermostat is expected to be closed. When a closure-malfunction occurs, the coolant temperature becomes considerably different from that normally expected for times at which the thermostat is expected to be opened. Alternatively, the malfunction may be detected from the difference between the engine side coolant temperature and the radiator side coolant temperature.