The cooling apparatus of the engine sets the first shut-off valve to the closed position, sets the second shut-off valve to the open position, and performs the opposite flow connection operation to supplies the cooling water directly to the cylinder block water passage from the cylinder head water passage without flowing the cooling water through the radiator and to the heat exchanger from the cylinder head water block when the engine temperature is lower than the completely-warmed temperature, and the cooling water is requested to be supplied to the heat exchanger.