An apparatus for purifying an exhaust gas for an internal combustion engine having units for the adsorption of HC gases in the exhaust gas, arranged in an exhaust manifold of the engine facing corresponding exhaust ports in such a manner that straight flows from the corresponding exhaust ports contact the corresponding adsorption units. A controller receives a signal indicating a temperature of one of a catalytic converter and the adsorption units. The controller sends a signal to a cooling apparatus, such as a fan, indicating a desired on condition of the cooling apparatus based on the temperature signal. The cooling apparatus is provided for generating air flows for cooling the adsorption units. The fan is operated when the engine, in a cold state, is started. The operation of the fan is continued until a catalytic converter arranged downstream from the adsorption units becomes effective, i.e. the temperature of the catalytic converter is higher than a predetermined value.