A cooling apparatus for cooling an internal combustion engine of a motorcycle, the cooling apparatus comprising: a cooling passage provided in the internal combustion engine and including an inlet through which cooling water flows in, and an outlet through which the cooling water flows out; a water pump including a discharge port through which the cooling water is discharged, and a suction port through which the cooling water is drawn in; a radiator including an inlet through which the cooling water flows in, and an outlet through which the cooling water flows out; a first passage connected to the discharge port of the water pump and the inlet of the cooling passage; a second passage connected to the outlet of the cooling passage and the inlet of the radiator; a third passage connected to the outlet of the radiator and the suction port of the water pump; an oil cooler passage including a first end portion connected to the second passage and a second end portion connected to the third passage, the oil cooler passage including an oil cooler; and a thermostat provided in a portion of the third passage between the outlet of the radiator and the second end portion, the thermostat being arranged to close when a temperature of the cooling water is lower than a reference temperature and to open when the temperature of the cooling water is equal to or higher than the reference temperature, wherein the thermostat includes a thermostat case provided with a first inlet (59i1), a second inlet (59i2), an outlet, and a valve body contained inside the thermostat case to open and close communication between the first inlet (59i1) and the outlet; the third passage includes an upstream passage connected to the outlet of the radiator and the first inlet (59i1) of the thermostat case, and a downstream passage connected to the outlet of the thermostat case and the suction port of the water pump; the oil cooler passage includes a downstream passage including an end portion connected to the oil cooler, and an end portion connected to the second inlet (59i2) of the thermostat case and defining the second end portion; and the thermostat is arranged to shut off communication between the first inlet (59i1) and the outlet by the valve body and allow communication between the second inlet (59i2) and the outlet when the temperature of the cooling water is lower than the reference temperature, and to allow communication between the first inlet (59i1) and the outlet and allow communication between the second inlet (59i2) and the outlet when the temperature of the cooling water is equal to or higher than the reference temperature, and wherein the oil cooler passage has a flow passage cross-sectional area smaller than flow passage cross-sectional areas of each of the second passage and the third passage. a cooling passage provided in the internal combustion engine and including an inlet through which cooling water flows in, and an outlet through which the cooling water flows out; a water pump including a discharge port through which the cooling water is discharged, and a suction port through which the cooling water is drawn in; a radiator including an inlet through which the cooling water flows in, and an outlet through which the cooling water flows out; a first passage connected to the discharge port of the water pump and the inlet of the cooling passage; a second passage connected to the outlet of the cooling passage and the inlet of the radiator; a third passage connected to the outlet of the radiator and the suction port of the water pump; an oil cooler passage including a first end portion connected to the second passage and a second end portion connected to the third passage, the oil cooler passage including an oil cooler; and a thermostat provided in a portion of the third passage between the outlet of the radiator and the second end portion, the thermostat being arranged to close when a temperature of the cooling water is lower than a reference temperature and to open when the temperature of the cooling water is equal to or higher than the reference temperature, wherein the thermostat includes a thermostat case provided with a first inlet (59i1), a second inlet (59i2), an outlet, and a valve body contained inside the thermostat case to open and close communication between the first inlet (59i1) and the outlet; the third passage includes an upstream passage connected to the outlet of the radiator and the first inlet (59i1) of the thermostat case, and a downstream passage connected to the outlet of the thermostat case and the suction port of the water pump; the oil cooler passage includes a downstream passage including an end portion connected to the oil cooler, and an end portion connected to the second inlet (59i2) of the thermostat case and defining the second end portion; and the thermostat is arranged to shut off communication between the first inlet (59i1) and the outlet by the valve body and allow communication between the second inlet (59i2) and the outlet when the temperature of the cooling water is lower than the reference temperature, and to allow communication between the first inlet (59i1) and the outlet and allow communication between the second inlet (59i2) and the outlet when the temperature of the cooling water is equal to or higher than the reference temperature, and wherein the oil cooler passage has a flow passage cross-sectional area smaller than flow passage cross-sectional areas of each of the second passage and the third passage.