The apparatus obtains a decreasing gradient of the maximum oxygen storing quantity Cmax with respect to the increase in an air flow rate Ga on the basis of two maximum oxygen storing quantities Cmax, which are respectively estimated in case where the air flow rate Ga assumes two different values, and the two values of the air flow rate Ga. As the deterioration degree of the catalyst unit 53 increases, the decreasing gradient increases. When the deterioration degree of the catalyst unit 53 is the same, the decreasing gradient tends to be the same, even if the degree of the response of the change in the output from a downstream air-fuel-ratio sensor 67 varies. Accordingly, the obtained decreasing gradient can be a value correctly indicating the deterioration degree of the catalyst unit 53.