A comprehensive study of structure, phase stability, electronic, vibrational and thermodynamic properties of LiMgSb and LiZnSb compounds is performed by carrying out First-principles calculations within densityfunctional theory using the full potential linearized augmented plane wave (FP-LAPW) combined with the pseudo-potential method. The generalized-gradient approximation (GGA) and the local density approximation (LDA) are chosen for the exchange-correlation energy. The Engel-Vosko (EV-GGA) formalism and mBJ approach are applied for the electronic properties. By exchanging the position of the atoms in the cubic structure, three phases (alpha, beta, gamma) are composed. Under ambient conditions, LiMgSb is stable in the cubic a structure however LiZnSb in the wurtzite one. At high pressure, these compounds undergo a structural phase transition from the cubic alpha to cubic gamma phase for LiMgSb and from the wurtzite to cubic gamma phase for LiZnSb, and the transition pressures were calculated. The quasi-harmonic Debye model, in which the phononic effects are predicted, is applied to the investigate of the thermodynamic properties. The temperature effect on the volume V(T), bulk modulus, B(T), thermal expansion coefficient alpha(T), specific heats (C-V and C-p) and Debye temperatures Theta(D), in different pressure and temperature range from 0 to 1000 K are investigated. Our results are in good agreement with previous theoretical calculations and the available experimental data. The band structure, density of states (DOS), and phonon dispersion curves have been obtained and analyzed.