A control device is installed in the vehicle that is able to execute a fuel cut that stops fuel supply to an engine in a state in which the engine is rotating. In a case where there is a request for the fuel cut while there is a heating request in which heating of a vehicle cabin is performed using heat of an engine coolant, when a blowout port mode of air conditioning air is set to a defroster mode or a bi-level mode, the control device prohibits the fuel cut.