This study develops a systematic density functional theory alongside on-site Coulomb interaction correction (DFT + U) and ab initio atomistic thermodynamics approachs for ternary (or mixed transitional metal oxides), expressed in three reservoirs. As a case study, among notable multiple metal oxides, synthesized CoCu2O3 exhibits favourable properties towards applications in solar, thermal and catalytic processes. This progressive contribution applies DFT + U and atomistic thermodynamic approaches to examine the structure and relative stability of CoCu2O3 surfaces. Twenty-five surfaces along the [001], [010], [100], [011], [101], [110] and [111] low-Miller-indices, with varying surface-termination configurations were selected in this study. The results portray satisfactory geometrical parameters for bulk CoCu2O3, and a band gap of 1.25 e V. Furthermore, we clarified the stoichiometrically balanced inverted (010)CoCuO, and the non-stoichiometric (001)CuOCu, (001)CoOCo, (110)OCoO and (110)CoOCu surface terminations as the most stable configurations, out of which, the (001)CuOCu shows the optimum stability in ambient conditions. The systematic approach applied in this study should prove instrumental for the analysis of other 3-element multicomponent systems. To the best of our knowledge, the present study is the first to report DFT + U analysis to any 3-multicompnent systems with two of them requires inclusion of U treatment (i.e., f- and d- orbitals) in the electronic structure calculations.