Effective thermal regulation is increasingly vital in numerous technological applications. The recent progress on topological diffusion in thermal materials offers promising potential. A distinct feature of diffusive systems is that their anti-Hermitian characteristics ensure purely imaginary eigenvalues representing decay rates, where higher decay rates imply better heat dissipation efficiency. Although research on higher-order topology has been extensive across various systems, investigations specifically of diffusive systems are just beginning. In particular, the few studies on the corner states are limited in their controllability. In this Letter, inspired by the concepts of the atomic $p$ orbital and higher-orbital topology, we introduce multipolar thermal modes in the domain wall between two thermal kagome lattices to achieve multiple corner states. Diverging from traditional research on kagome lattices, we discovered higher-decay-rate corner states above the third band and linked their properties to multipolar modes localized at the domain wall. This approach not only enables higher-decay-rate corner states but also explores new dimensions of control over them. We theoretically identify and experimentally observe various corner states in a two-dimensional diffusive system. It is also demonstrated that simple thermal pulses can trigger transitions between different corner states, thereby dynamically enhancing the decay rate. This study further reveals the potential of topological approaches in heat dissipation, paving the way for the development of thermal management and related fields.

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• Edge states
• Geometric & topological phases
• Heat transfer
• Topological materials