Data obtained in the present work and available publications on combustion of cyclotetramethylene tetranitramine (HMX) at different initial temperatures are analyzed. The temperature sensitivity of the HMX burning rate is demonstrated to increase with increasing initial temperature at pressures of 0.1 to 10 MPa, which is typical for combustion of substances with the leading reaction in the condensed phase (c-phase model). Experimental values of the temperature sensitivity of the burning rate in the pressure interval between 0.1 and 1 MPa are higher than the values predicted by the c-phase model, but this fact indicates the transition of the combustion process to another regime rather than the combustion instability in this area. The flame structure of burning HMX with different additives is studied with the help of thin tungsten-rhenium thermocouples in the pressure range from 0.025 to 1 MPa. The gas-phase flame is found to ignite in an inductive mode, at least up to a pressure of 1 MPa. The surface temperature is obtained as a function of pressure on the basis of experimental data in a wide range of pressures: ln p = −14,092/T + 21.72 (p in atm). Two possible reasons for the oscillatory regime of HMX combustion observed at atmospheric pressure are proposed: the emergence of resonance phenomena during combustion of an inhomogeneous gas mixture in the tube and the lack of correspondence between the chemical reaction rate in the gas phase at the instant of the resonance and its energy capabilities, which do not allow a necessary HMX gasification rate to be ensured. A mechanism of HMX combustion is proposed, which offers an adequate description in a wide range of pressures up to 10 MPa. The mechanism is based on the leading role of HMX decomposition in the melt at the surface temperature.