The structural, elastic, electronic, optical and thermodynamic properties of the sodium polonide Na2Po compound have been studied through the full potential linearized augmented plane wave plus local orbitals (FP-LAPW + lo) and tight-binding linear muffin-tin orbital (TB-LMTO) methods. The exchange–correlation potential was treated within the local density approximation for the TB-LMTO calculations and within the generalized gradient approximation for the FP-LAPW + lo calculations. In addition, Tran and Blaha-modified Becke–Johnson (TB-mBJ) potential and Engel–Vosko generalized gradient approximation were used for the electronic and optical properties. Ground state properties such as the equilibrium lattice constant, bulk modulus and its pressure derivative were calculated and compared with available data. The single-crystal and polycrystalline elastic constants of the considered compound were calculated via the total energy versus strain in the framework of the FP-LAPW + lo approach. The calculated electronic structure reveals that Na2Po is a direct band gap semiconductor. The frequency-dependent dielectric function, refractive index, extinction coefficient, reflectivity coefficient and electron energy loss function spectra are calculated for a wide energy range. The variations of the lattice constant, bulk modulus, heat capacity, volume expansion coefficient and Debye temperature with temperature and pressure were calculated successfully using the FP-LAPW + lo method in combination with the quasi-harmonic Debye model.