<p>The scaling law of transient two-layer electroosmotic flow with slip-dependent zeta potential in cylindrical and parallel-plate microchannels is derived using the eigenfunction expansion method and asymptotic matching for small zeta potential. A universal formula is derived by capturing the essence of transient response of two-layer flow at given ranges of parameters. The scaled flow rates collapse onto a single universality curve expressed by the universal formula no matter what parameters or geometries are considered. The universality underlying the temporal evolution of two-layer flow is revealed, which is that the steady flow rate is related to the intrinsic properties of two-layer flow and external forces, while the transient flow rate is primarily determined by the intrinsic properties. The rapid prediction of transient flow rate at arbitrary parameters is allowed without obtaining the velocity field, as long as the steady solution and first eigenvalue of two-layer system are known.</p>