Many resonant structures in the microwave region need damping of parasitic or higher order modes in order to optimise their performance. In cavities or other systems operating in accelerating structures, for example, the mitigation of spurious resonance effects is indispensable to achieve high quality particle beams. We present here a parametric study on the absorption properties of a multilayered metamaterial having the shape of a truncated pyramid to be used as light, small volume damper in a microwave cavity. Full wave simulations have been performed for a single absorber varying its size and number of layers and for a linear array changing the distance between the single elements. A case study is proposed, where a single pyramid is inserted in a real pillbox cavity. Measurements of the cavity response without and with the metamaterial absorber are presented and compared with numerical results.