The structural, elastic, electronic, and thermodynamic properties of thermoelectric material Mg Ag Sb in γ, β, α phases are studied with first-principles calculations based on density functional theory. The optimized lattice constants accord well with the experimental data. According to the calculated total energy of the three phases, the phase transition order is determined from α to γ phase with cooling, which is in agreement with the experimental result. The physical properties such as elastic constants, bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, and anisotropy factor are also discussed and analyzed, which indicates that the three structures are mechanically stable and each has a ductile feature. The Debye temperature is deduced from the elastic properties. The total density of states(TDOS) and partial density of states(PDOS) of the three phases are investigated. The TDOS results show that the γ phase is most stable with a pseudogap near the Fermi level, and the PDOS analysis indicates that the conduction band of the three phases is composed mostly of Mg-3s,Ag-4d, and Sb-5p. In addition, the changes of the free energy, entropy, specific heat, thermal expansion of γ-MgAgSb with temperature are obtained successfully. The obtained results above are important parameters for further experimental and theoretical tuning of doped MgAgSb as a thermoelectric material at high temperature. The structural, elastic, electronic, and thermodynamic properties of thermoelectric material Mg Ag Sb in γ, β, α phases are studied with first-principles calculations based on density functional theory. The optimized lattice constants accord well with the experimental data. According to the the calculated total energy of the three phases, the phase transition order is determined from α to γ ​​phase with cooling, which is in agreement with the experimental result. The physical properties of such elastic constants, bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio , and anisotropy factor are also discussed and analyzed, which indicates that the three structures are mechanically stable and each has a ductile feature. The total density of states (TDOS) and partial density of states PDOS) of the three phases are investigated. The TDOS results show that the γ phase is most stable with a pseudogap near the Fermi level, an d the PDOS analysis indicates that the conduction band of the three phases is mostly composed of Mg-3s, Ag-4d, and Sb-5p. In addition, the changes of the free energy, entropy, specific heat, thermal expansion of γ- MgAgb with temperature are obtained successfully. The obtained results are above parameters for further experimental and theoretical tuning of doped MgAgSb as a thermoelectric material at high temperature.