The potential energy surface of a model discotic liquid crystal is investigated across mesophases as a function of temperature by characterizing the local potential energy minima. The average depth of the minima sampled by the system gradually grows as orientational order increases through the discotic-nematic phase upon cooling, while it remains fairly constant in the isotropic phase for isochoric temperature variation. The high-temperature Arrhenius behavior of the single-particle orientational correlation times is found to break down at a temperature that marks the onset of exploration of deeper potential energy minima. The structural features of the minima reveal an interplay between orientational and translational order, in particular, when the parent phase is discotic-nematic. The local minima then exhibit short-range columns that tend to have local hexagonal packing. The present study and recent work on calamitic liquid crystals [D. Chakrabarti and B. Bagchi, Proc. Natl. Acad. Sci. U.S.A. 103, 7217 (2006)] together reveal a striking similarity between thermotropic liquid crystals and supercooled liquids in the exploration of the energy landscape and the breakdown of Arrhenius behavior for relaxation times.