Cubic perovskite oxide BaSnO₃ has attracted increasing attention because of its exceptional optoelectronic properties and promising device applications. However, its composition of the most stable surface and relative thermodynamic stability of the different surface terminations remain controversial. Here we report a comprehensive computational and theoretical study of various (001) surface structures of BaSnO₃ using first-principles electronic structure calculations. Our calculations show that the stoichiometric BaO surface termination has a slightly lower surface energy than SnO₂ surface termination, although their values are comparable to each other and about one-half of the cleavage energy of BaSnO₃. The calculated phase stability diagrams of the BaSnO₃ (001) surface show that BaO surface termination is thermodynamically preferred to other surface terminations in the allowed range of BaSnO₃ without forming secondary phases. Our calculations of cleavage energy reveal that eliminating the surface layer of BaO$\cdot$H₂O requires less energy compared to the surface layer of SnO₂ $\cdot$H₂O. These results explain the experimental findings why BaSnO₃ films have a Ba-excess surface after oxygen annealing while SnO₂ surface termination can only be achieved upon water leaching. This work may offer some guidelines for achieving precise control over the surface terminations of BaSnO₃.