We utilize dynamic light scattering (DLS) and passive microrheology to examine the phase behavior of a supramolecular polymer at very high pressures. The monomer, 2,4-bis(2ethylhexylureido)toluene (EHUT), self-assembles into supramolecular polymeric structures in the non-polar solvent cyclohexane, by means of hydrogen bonding. By varying concentration and temperature at atmospheric pressure, the formation of viscoelastic network (at lower temperatures) and predominantly viscous phases, based on self-assembled tube and filament structures respectively, has been established. The associated changes in the rheological properties have been attributed to a structural thickness transition. Here, we investigate the effects of pressure variation, from atmospheric up to 1 kbar. We construct a temperature-pressure diagram that reveals the predominance of the viscoelastic network phase at high pressures. The transition from viscoelastic network organization of the tubes to a weaker viscous-dominated