Two structural phase transitions are investigated in highly strained BiFeO₃ thin films as a function of film thickness and temperature via synchrotron x-ray diffraction. Both transition temperatures (upon heating: monoclinic M_C to monoclinic M_A to tetragonal) decrease as the film becomes thinner. A film-substrate interface layer, evidenced by half-order peaks, contributes to this behavior, but at larger thicknesses (above a few nanometers), the temperature dependence results from electrostatic considerations akin to size effects in ferroelectric phase transitions, but observed here for structural phase transitions within the ferroelectric phase. For ultra-thin films, the tetragonal structure is stable to low temperatures.