Energy decay for the Klein-Gordon equation with highly oscillating damping

We consider the free Klein-Gordon equation with periodic damping. We show on this simple model that if the usual geometric condition holds then the decay of the energy is uniform with respect to the oscillations of the damping, and in particular the size of the derivatives do not play any role. We also show that without geometric condition the polynomial decay of the energy is even slightly better for a highly oscillating damping. To prove these estimates we provide a parameter dependent version of well known results of semigroup theory