•Magnetically disordered phase is suppressed under pressure.•The Néel temperature increases substantially under pressure.•The mechanism of next nearest neighbor superexchange interactions of A-O-O-A type is proposed. The crystal and magnetic structures of the frustrated Co3O4 spinel oxide have been studied by means of neutron diffraction at high pressures up to 8.7 GPa in the temperature range 5–300 K. At ambient pressure, a coexistence of the long range ordered antiferromagnetic (AFM) phase and short range disordered phase below TN = 33 K was observed. At pressures above 2 GPa, a suppression of the magnetically disordered phase and stabilization of the AFM phase, characterized by rapid increase of the Néel temperature in 1.5 times from 33 to 51 K with a pressure coefficient dTN/dP = 2.1 K/GPa occur. The observed phenomena are analyzed in terms of the competing nearest neighbor and next nearest neighbor magnetic interactions. In addition to the A-O-B-O-A nearest neighbor superexchange, an important role of the A-O-O-A next nearest neighbor superexchange interactions in magnetic behavior of Co3O4 is pointed out.