To investigate the stability evolution of mountainous slopes in karst regions, this study considers the deterioration of pipelines along the rear edge and sliding surface, as well as the weakening effects of water on structural planes. Taking planar sliding in karst mountains as the research object, two geological models and corresponding mechanical models are established, including unconnected and connected karst conduits. Based on cusp catastrophe theory, the stability coefficient (FOS) and critical stability coefficient (FOS^*) of karst slopes under pipeline flow conditions are derived, and an energy-based criterion for abrupt instability is proposed. Results show that the displacement relationship between equilibrium and critical instability points can effectively determine the onset of slope failure. When conduits are unconnected, the presence of an additional stiffness coefficient k in the formulations of FOS and FOS^* reveals that karst conduits influence slope stability. The stability coefficient is closely related to both k and pipeline deterioration coefficient m, showing their controlling roles in stability evolution. A case study of the Jiwei Mountain landslide verifies that the proposed energy criterion based on cusp catastrophe theory more accurately evaluates the stability of karst slopes with complex sliding surfaces, complementing the results of traditional limit equilibrium analysis.