The catalytic wet air oxidation of p-coumaric acid, a representative substrate of biorecalcitrant phenolic compounds typically found in olive oil processing wastewater, was carried out under mild conditions (in air at T = 353 K and P = 2 MPa) with activated carbon (AC) and multiwalled carbon nanotubes (CNTs). The influence of the CNT textural and surface chemistry modification, by oxidation with a nitric/sulfuric acid mixture, on the catalytic behavior was evaluated. AC and unoxidized CNTs showed the highest activity toward the degradation of p-coumaric acid and the highest efficiency toward total organic carbon (TOC) removal, which mainly occurs through mineralization to CO2. As a result of oxidation with an acid mixture, the stability of the CNTs was dramatically worsened. From a comparison with literature data, it was concluded that the most efficient AC and unoxidized CNTs catalysts are, so far, very promising systems for p-coumaric acid degradation under mild conditions.