A rotational speed control apparatus for an engine, the engine configured to drive a compressor that compresses a cooling medium for air conditioning, the rotational speed control apparatus comprising: an electronic control unit configured to control engine rotational speed during idle operation in a normal period and a changeover transition period, the changeover transition period being a predetermined period from a start of a changeover between a driven state of the compressor and a stopped state of the compressor, and the normal period being other than the changeover transition period, the electronic control unit being configured in the normal period to: (a) correct a required value of a torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate a load torque of the compressor, and (c) correct the required value of the torque of the engine in accordance with the value of the calculated value of the load torque, characterized in that the electronic control unit is further configured only in the changeover transition period to: (a) correct the required value of the torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate the load torque of the compressor; (c) correct the calculated value of the load torque through feedback in accordance with a deviation between the rotational speed of the engine and the target rotational speed, and (d) correct the required value of the torque of the engine in accordance with the value of the corrected value of the load torque. an electronic control unit configured to control engine rotational speed during idle operation in a normal period and a changeover transition period, the changeover transition period being a predetermined period from a start of a changeover between a driven state of the compressor and a stopped state of the compressor, and the normal period being other than the changeover transition period, the electronic control unit being configured in the normal period to: (a) correct a required value of a torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate a load torque of the compressor, and (c) correct the required value of the torque of the engine in accordance with the value of the calculated value of the load torque, characterized in that (a) correct a required value of a torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate a load torque of the compressor, and (c) correct the required value of the torque of the engine in accordance with the value of the calculated value of the load torque, characterized in that the electronic control unit is further configured only in the changeover transition period to: (a) correct the required value of the torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate the load torque of the compressor; (c) correct the calculated value of the load torque through feedback in accordance with a deviation between the rotational speed of the engine and the target rotational speed, and (d) correct the required value of the torque of the engine in accordance with the value of the corrected value of the load torque. (a) correct the required value of the torque of the engine through feedback in accordance with a deviation between a rotational speed of the engine during idle operation and a target rotational speed; (b) calculate the load torque of the compressor; (c) correct the calculated value of the load torque through feedback in accordance with a deviation between the rotational speed of the engine and the target rotational speed, and (d) correct the required value of the torque of the engine in accordance with the value of the corrected value of the load torque.