The direct conversion of methane into alcohol is a promising approach for achieving a low-carbon future, yet it remains a major challenge. In this study, we utilize density functional theory to explore the potential of the (CoCrFeMnNi)3O4 (CCFMN) high entropy oxide (HEO) for electrochemical oxidation of methane to methanol and ethanol, alongside their competition with CO2 production. Our primary focus in this study is on thermodynamics, enabling a prompt analysis of the catalyst's potential, with the calculation of electrochemical barriers falling beyond our scope. Among all potential active sites within the CCFMN HEO, we identify Co as the most active site for methane activation when using carbonate ions as oxidants. This results in methanol production with a limiting potential of 1.4 VCHE, and ethanol and CO2 productions with a limiting potential of 1.2 VCHE. Additionally, our findings suggest that the occupied p-band center of O* on the CCFMNO HEO is a potential descriptor for identifying the most active site within the CCFMNO HEO. Overall, our results indicate that the CCFMN HEO holds promise as catalysts for methane oxidation to alcohols, employing carbonate ions as oxidants