Diffusive nature of thermal transportation fundamentally restricts topological characteristics due to the absence of a sufficient parametric space with complex dimensionalities. Here, we create an orthogonal advection space with two advective pairs to reveal the unexplored topological transitions in thermal material. We demonstrate four types of configurable thermal phases, including the nontrivial dynamic-equilibrium distribution, nonchiral steplike π-phase transition, and another two trivial profiles related to the anti-parity-time symmetry nature. Our findings provide a recipe for realizing a topologically robust thermal system under arbitrary perturbations.