Supercapacitors prepared from chemically modified and vacuum filtered buckypapers were studied. The aim was to evaluate how its pore structure impacts the specific capacitance, energy, and power density in different electrolytes. The specific capacitance varies in a linear fashion with the specific surface area for nanotubes modified by the means of catalytic, low-temperature partial catalytic oxidation using cobalt oxide nanoparticles decorating the nanotubes. In contrast, electrodes composed of nanotubes preactivated in CO2 demonstrated only a minor increase in their specific capacitance, despite the observed significant increase in specific surface area. The radically improved surface area was a result of emergence and deposition of soot on the nanotubes during the activation process, as revealed by transmission electron microscopy. Among six different types of electrode materials, the CoOx decorated materials proved to have the highest specific capacitance (∼25 F/g in aqueous KOH and ∼15 F/g in triethylsulfonium bis(trifluoromethylsulfonyl)imide ionic liquid). Thus, highly structured carbon nanotubes giving rise to energy and power storage densities comparable with commercial and other multiwalled carbon nanotube based electric double-layer capacitor devices were obtained.