A cooling device for an internal combustion engine, comprising: a HT cooling system that mainly cools a cylinder block of the internal combustion engine; a LT cooling system comprising a cooling pump and a fan, that mainly cools a periphery of an intake port, the LT cooling system and the HT cooling system having cooling medium flow passages independent of each other; and an electronic control unit configured to: when a HT temperature being the temperature of a HT cooling medium flowing in the HT cooling system has reached a HT determination value, control an operation state of the HT cooling system to start cooling for maintaining the HT temperature at a HT target temperature, the cooling device being characterized in that the electronic control unit is further configured to: LT temperature being the temperature of a LT cooling medium flowing in the LT cooling system has reached a LT determination value while the cooling pump and the fan are stopped, start a LT cooling control under a condition where neither early warm-up of the internal combustion engine nor knock suppression are required, the LT cooling control being a control in which at least the cooling pump is driven for maintaining the LT temperature at a LT target temperature, an early warm up requirement corresponding to when a heater in a cabin, early warm up of a catalyst for exhaust gas purification, or EGR introduction are required, and start the LT cooling control when the HT temperature has reached the HT determination value when the early warm-up of the internal combustion engine is required. a HT cooling system that mainly cools a cylinder block of the internal combustion engine; a LT cooling system comprising a cooling pump and a fan, that mainly cools a periphery of an intake port, the LT cooling system and the HT cooling system having cooling medium flow passages independent of each other; and an electronic control unit configured to: when a HT temperature being the temperature of a HT cooling medium flowing in the HT cooling system has reached a HT determination value, control an operation state of the HT cooling system to start cooling for maintaining the HT temperature at a HT target temperature, the cooling device being characterized in that the electronic control unit is further configured to: LT temperature being the temperature of a LT cooling medium flowing in the LT cooling system has reached a LT determination value while the cooling pump and the fan are stopped, start a LT cooling control under a condition where neither early warm-up of the internal combustion engine nor knock suppression are required, the LT cooling control being a control in which at least the cooling pump is driven for maintaining the LT temperature at a LT target temperature, an early warm up requirement corresponding to when a heater in a cabin, early warm up of a catalyst for exhaust gas purification, or EGR introduction are required, and start the LT cooling control when the HT temperature has reached the HT determination value when the early warm-up of the internal combustion engine is required. when a HT temperature being the temperature of a HT cooling medium flowing in the HT cooling system has reached a HT determination value, control an operation state of the HT cooling system to start cooling for maintaining the HT temperature at a HT target temperature, the cooling device being characterized in that the electronic control unit is further configured to: LT temperature being the temperature of a LT cooling medium flowing in the LT cooling system has reached a LT determination value while the cooling pump and the fan are stopped, start a LT cooling control under a condition where neither early warm-up of the internal combustion engine nor knock suppression are required, the LT cooling control being a control in which at least the cooling pump is driven for maintaining the LT temperature at a LT target temperature, an early warm up requirement corresponding to when a heater in a cabin, early warm up of a catalyst for exhaust gas purification, or EGR introduction are required, and start the LT cooling control when the HT temperature has reached the HT determination value when the early warm-up of the internal combustion engine is required.