A coast stop vehicle includes a hydraulic pressure supplying unit which supplies a hydraulic pressure to the transmission while the engine is stopped, a coast stop start condition judging unit which judges whether or not a coast stop start condition holds, a coast stop control unit which stops the engine when the coast stop start condition is judged to hold by the coast stop start condition judging unit and starts the engine when receiving an acceleration request from a driver during the coast stop, a transmission control unit which causes a downshift of the transmission when receiving the acceleration request from the driver during the coast stop, and a hydraulic control unit which supplies the hydraulic pressure to an after-shift frictional engagement element which realizes a gear position after the downshift of the transmission during the coast stop.