Metal-organic frameworks (MOFs) are microporous materials with huge potential for chemical processes. Structural collapse at high pressure, and transitions to liquid states at high temperature, have recently been observed in the zeolitic imidazolate framework (ZIF) family of MOFs. Here, we show that simultaneous high-pressure and high-temperature conditions result in complex behaviour in ZIF-62 and ZIF-4, with distinct high- and low-density amorphous phases occurring over different regions of the pressure-temperature phase diagram. In situ powder X-ray diffraction, Raman spectroscopy and optical microscopy reveal that the stability of the liquid MOF state expands substantially towards lower temperatures at intermediate, industrially achievable pressures and first-principles molecular dynamics show that softening of the framework coordination with pressure makes melting thermodynamically easier. Furthermore, the MOF glass formed by melt quenching the high-temperature liquid possesses permanent, accessible porosity. Our results thus imply a route to the synthesis of functional MOF glasses at low temperatures, avoiding decomposition on heating at ambient pressure.