We study cobalt/platinum (Co/Pt) superlattices by means of the ferromagnetic resonance spectroscopy in the frequency domain. The magnetization easy-axis of superlattices, which lies in a plane of the film with Co layers of 1.5 and 1.2 nm thickness, changes to the direction perpendicular to the film surface with Co layers of 0.9 and 0.6 nm thickness. As the Co layer thickness decreases from 0.9 to 0.6 nm, the Gilbert damping parameter significantly increases while the $g$-value is almost constant. As the Pt layer thickness decreases from 0.6 to 0.3 nm in the superlattices with 0.9 nm thick Co layers, the damping parameter slightly decreases. The present study demonstrates that the Co/Pt superlattices with 0.9 nm thick Co layers and 0.3 nm thick Pt layers are suitable for time-varying magnetic metamaterials realizing a larger magnetic permeability modulation.