A hybrid capacitor includes an electrically non-conductive rigid or semi-rigid porous honeycomb structure having cells extending along a common direction, the cells having a plurality of cross-sectional shapes. The honeycomb structure is desirably formed of a material that is stable at temperatures of 300° or more, such that high temperature processing can be used to help ensure high purity of the final product. The material of the structure may desirably be an oxide or non-oxide ceramic, such as cordierite, silicon nitride, alumina, aluminum titanate, zircon, glass, or glass-ceramic. The plurality of shapes of the cells includes larger shapes in which cells are disposed non-galvanic electrodes, with galvanic electrodes disposed in cells of other shapes.