We present theoretical investigations of pressure- and temperature-driven phase transitions in HgTequantum
wells grown on a CdTe buffer. Using the eight-band k·p Hamiltonian we calculate evolution of energy-band structure at different quantum well widths with hydrostatic pressure up to 20 kbars and temperature ranging up to 300 K. In particular, we show that, in addition to temperature, tuning of hydrostatic pressure allows us to drive transitions between semimetal, band insulator, and topological insulator phases. Our realistic band-structure calculations reveal that the band inversion under hydrostatic pressure and temperature may be accompanied by nonlocal overlapping between conduction and valence bands. The pressure and temperature phase diagrams are presented.