  The integration of topological concepts into electronic energy band theory
has been a transformative development in condensed matter physics. Since then,
this paradigm has broadened its reach, extending to a variety of physical
systems, including open ones. In this study, we employ analogues of the
generalized $n$-dimensional Su-Schrieffer-Heeger model, a cornerstone in
understanding topological insulators and higher-order topological states, to
unveil a dimensional hierarchy of topological states within thermal diffusive
networks. Unlike their electronic counterparts, the topological states in these
networks are characterized by confined temperature profiles of dimension
$(n-d)$ with constant diffusive rates, where $n$ represents the system's
dimension and $d$ is the order of the topological state. Our findings
demonstrate the existence of topological corner states in thermal diffusive
systems up to $n=3$, along with surface and hinge states. We also identify and
discuss an intermediate-order topological phase in the case $n=3$,
characterized by the presence of hinge states but the absence of corner states.
Furthermore, our work delves into the influence of chiral symmetry in these
thermal networks, particularly focusing on topological thermal states with a
near-zero diffusion rate. This research lays the foundation for advanced
thermal management strategies that utilize topological states in multiple
dimensions.
