The equation for specific capacitance is: Cs = 1/2 * CV area/(scan rate * voltage window * mass of material)

Since the current is a response to the potential sweep, and there is a direct proportion of Cs to area of CV, the higher current would give the higher specific capacitance.

If you are talking about current as in a charging current, CV uses a potential sweep so the closest thing would be the scan rate (mV/s). This is already accounted for in the equation so it you use a fast or slow scan rate you should arrive at the same Cs