Disclinations-topological defects ubiquitously existing in various materials-can reveal the intrinsic band topology of the hosting material through the bulk-disclination correspondence. In low-dimensional materials and nanostructure such as graphene and fullerenes, disclinations yield curved surfaces and emergent non-Euclidean geometries that are crucial in understanding the properties of these materials. However, the bulk-disclination correspondence has never been studied in non-Euclidean geometry, nor in systems with p-orbital physics. Here, by creating p-orbital topological acoustic metamaterials with disclination-induced conic and hyperbolic surfaces, we demonstrate the rich emergent bound states arising from the interplay among the real-space geometry, the bulk band topology, and the p-orbital physics. This phenomenon is confirmed by clear experimental evidence that is consistent with theory and simulations. Our experiment paves the way toward topological phenomena in non-Euclidean geometries and will stimulate interesting research on, e.g., topological phenomena for electrons in nanomaterials with curved surfaces.