Molecular dynamics simulations of a methane pair in water are used to calculate the thermodynamic properties of the hydrophobic interaction as a function of pressure. Pressure is found to decrease the tendency to form methane aggregates. The entropic contribution to the free energy, which at atmospheric pressure greatly stabilizes aggregation, is highly pressure dependent. As the pressure increases, the entropic stabilization steadily decreases until, at 7 kbar, the entropy of the contact pair is equal to the entropy of the solvent separated pair. The heat capacity change between the contact and solvent separated Fair is shown to be large and positive at 1 atm, as is characteristic of hydrophobic processes. At higher pressures, the heat capacity change is zero, indicating that two of the significant properties of the hydrophobic effect, the large entropy decrease and heat capacity increase are lost at high pressures. The free energy, volume and entropy changes are consistent with the corresponding changes for the pressure denaturation of proteins.