The structural stability, equation of state, and thermal expansion behavior of nanocrystalline cubic HfO2, an ultra-high-temperature ceramic, have been investigated using X-ray diffraction at extreme conditions of pressures and temperatures. High-pressure studies show that the cubic structure is stable up to 26.2 GPa, while the high-temperature studies show the stability of the cubic structure up to 600 degrees C. The Rietveld structure refinement of the high-pressure data reveals the progressive transition of secondary monoclinic phase to the cubic phase at higher pressures. The phase progression is accompanied by incompressibility along the b axis and a large compressibility along the c axis of the monoclinic structure. The second-order Birch-Murnaghan equation of state fit to the unit cell volume data yielded a bulk modulus of 242(16) GPa for the cubic structure. A linear thermal expansion value of alpha(a(c)) = 8.80(15) x 10(-6)degrees C-1 and a volume thermal expansion value of alpha(v) = 26.5(4) x 10(-6)degrees C-1 have been determined from the in situ high-temperature X-ray diffraction studies. The results are discussed by comparing with the high-pressure and high-temperature behavior of isostructural ZrO2. To the best of our knowledge, this is the first experimental report on the structural stability of cubic HfO2 at high pressures.