A cooling device for an engine, the cooling device comprising: a coolant circuit including a pump, a radiator, and a plurality of routes, the plurality of routes being configured such that a coolant flows from the pump through the inside of the engine and returns to the pump, the plurality of routes being branched at a branching position on a downstream side of the inside of the engine, the plurality of routes being each connected to the pump, the plurality of routes including a radiator route, the radiator route passing through the radiator; a multiway valve including a plurality of discharge ports, the discharge ports being provided at the branching position of the plurality of routes in the coolant circuit, the discharge ports being configured to discharge the coolant respectively to the plurality of routes, the discharge ports including a radiator port, the radiator port being a discharge port that is configured to discharge the coolant to the radiator route, the multiway valve being configured to switch between open and closed states of the discharge ports, the open and closed states of the discharge ports including a state in which all the discharge ports are closed; characterised in further comprising an electronic control unit configured to control the multiway valve such that the radiator port closes and at least one of the discharge ports, other than the radiator port, opens when an ignition switch is turned off. a coolant circuit including a pump, a radiator, and a plurality of routes, the plurality of routes being configured such that a coolant flows from the pump through the inside of the engine and returns to the pump, the plurality of routes being branched at a branching position on a downstream side of the inside of the engine, the plurality of routes being each connected to the pump, the plurality of routes including a radiator route, the radiator route passing through the radiator; the plurality of routes being configured such that a coolant flows from the pump through the inside of the engine and returns to the pump, the plurality of routes being branched at a branching position on a downstream side of the inside of the engine, the plurality of routes being each connected to the pump, the plurality of routes including a radiator route, the radiator route passing through the radiator; a multiway valve including a plurality of discharge ports, the discharge ports being provided at the branching position of the plurality of routes in the coolant circuit, the discharge ports being configured to discharge the coolant respectively to the plurality of routes, the discharge ports including a radiator port, the radiator port being a discharge port that is configured to discharge the coolant to the radiator route, the multiway valve being configured to switch between open and closed states of the discharge ports, the open and closed states of the discharge ports including a state in which all the discharge ports are closed; characterised in further comprising an electronic control unit configured to control the multiway valve such that the radiator port closes and at least one of the discharge ports, other than the radiator port, opens when an ignition switch is turned off. the discharge ports being provided at the branching position of the plurality of routes in the coolant circuit, the discharge ports being configured to discharge the coolant respectively to the plurality of routes, the discharge ports including a radiator port, the radiator port being a discharge port that is configured to discharge the coolant to the radiator route, the multiway valve being configured to switch between open and closed states of the discharge ports, the open and closed states of the discharge ports including a state in which all the discharge ports are closed; characterised in further comprising an electronic control unit configured to control the multiway valve such that the radiator port closes and at least one of the discharge ports, other than the radiator port, opens when an ignition switch is turned off.