Mo2Ga2C is a new MAX phase with a stacking Ga bilayer as well as possible unusual properties. To understand this unique MAX-phase structure and promote possible future applications, the structure, chemical bonding, mechanical and thermodynamic properties of Mo2Ga2C were investigated by first principles. Using the "bond stiffness" model, the strongest covalent bonding (1162 GPa) were formed between Mo and C atoms in Mo2Ga2C, while the weakest Ga-Ga (389 GPa) bonding were formed between two Ga-atomic layers, different from other typical MAX phases. Of interest, the ratio of the bond stiffness of the weakest bond to the strongest bond (0.33) was lower than 1/2, indicating the high damage tolerance and fracture toughness of Mo2Ga2C, which was confirmed by indentation without any cracks. The high-temperature heat capacity and thermal expansion of Mo2Ga2C were calculated in the framework of quasi-harmonic approximation from 0 K to 2000 K. Because of the metal-like electronic structure, the electronic excitation contribution became more significant with increasing temperature above 300 K.