An apparatus for a vehicle having an engine that is selectively controllable to stop automatically when a predetermined condition is satisfied, at least one exhaust gas sensor associated with an exhaust manifold of the engine for sensing a property of an exhaust gas and a heating device for heating the exhaust gas sensor, the apparatus comprising: environmental condition sensing means for sensing at least one environmental condition including at least one of atmospheric temperature and atmospheric pressure; and controller means, comprising: condensation occurrence temperature estimation means for estimating a condensation occurrence temperature based on water vapor partial pressure in the exhaust manifold and on the at least one environmental condition; and heating control means adapted for, when the engine is automatically stopped: - selectively heating the exhaust gas sensor to a desired activation temperature using the heating device if a temperature of the exhaust manifold is higher than or equal to the condensation occurrence temperature, and - selectively performing one of lowering heating performance of the heating device and stopping the heating performed by the heating device if the exhaust manifold temperature is lower than the condensation occurrence temperature, characterized in that the controller means is arranged to selectively determine the water vapor partial pressure in the exhaust manifold, and wherein the condensation occurrence temperature is set to a temperature at which the water vapor partial pressure is equal to or substantially equal to the saturated water vapor pressure, the controller means comprises an exhaust gas water vapor partial pressure calculation means arranged to selectively calculate the water vapor partial pressure of a gas that resides within the exhaust manifold when the engine is automatically stopped, the exhaust gas water vapor partial pressure calculation means is arranged to calculate, if the engine is stopped by cutting off the fuel supply when the engine is in an idle state, the water vapor partial pressure of the gas that resides within the exhaust manifold, based on the water vapor partial pressure of exhaust gas in the idle state and based on the water vapor partial pressure of an intake air that is exhausted to the exhaust manifold by rotation of the engine after the fuel supply is cut off environmental condition sensing means for sensing at least one environmental condition including at least one of atmospheric temperature and atmospheric pressure; and controller means, comprising: condensation occurrence temperature estimation means for estimating a condensation occurrence temperature based on water vapor partial pressure in the exhaust manifold and on the at least one environmental condition; and heating control means adapted for, when the engine is automatically stopped: - selectively heating the exhaust gas sensor to a desired activation temperature using the heating device if a temperature of the exhaust manifold is higher than or equal to the condensation occurrence temperature, and - selectively performing one of lowering heating performance of the heating device and stopping the heating performed by the heating device if the exhaust manifold temperature is lower than the condensation occurrence temperature, condensation occurrence temperature estimation means for estimating a condensation occurrence temperature based on water vapor partial pressure in the exhaust manifold and on the at least one environmental condition; and heating control means adapted for, when the engine is automatically stopped: - selectively heating the exhaust gas sensor to a desired activation temperature using the heating device if a temperature of the exhaust manifold is higher than or equal to the condensation occurrence temperature, and - selectively performing one of lowering heating performance of the heating device and stopping the heating performed by the heating device if the exhaust manifold temperature is lower than the condensation occurrence temperature, - selectively heating the exhaust gas sensor to a desired activation temperature using the heating device if a temperature of the exhaust manifold is higher than or equal to the condensation occurrence temperature, and - selectively performing one of lowering heating performance of the heating device and stopping the heating performed by the heating device if the exhaust manifold temperature is lower than the condensation occurrence temperature, characterized in that the controller means is arranged to selectively determine the water vapor partial pressure in the exhaust manifold, and wherein the condensation occurrence temperature is set to a temperature at which the water vapor partial pressure is equal to or substantially equal to the saturated water vapor pressure, the controller means comprises an exhaust gas water vapor partial pressure calculation means arranged to selectively calculate the water vapor partial pressure of a gas that resides within the exhaust manifold when the engine is automatically stopped, the exhaust gas water vapor partial pressure calculation means is arranged to calculate, if the engine is stopped by cutting off the fuel supply when the engine is in an idle state, the water vapor partial pressure of the gas that resides within the exhaust manifold, based on the water vapor partial pressure of exhaust gas in the idle state and based on the water vapor partial pressure of an intake air that is exhausted to the exhaust manifold by rotation of the engine after the fuel supply is cut off