Thermal fluctuations can destroy or stabilize ordered phases, but can they also generate topological bands and boundary modes? A central obstacle is that fluctuations reshape excitation spectra while simultaneously broadening them, potentially washing out topological gaps. Here we demonstrate an entropic mechanism for band topology in anharmonic mechanical lattices. Thermal fluctuations renormalize lattice interactions, inducing topological transitions and the emergence of topological edge states. We show, through the agreement between analytic theory and molecular dynamics simulations, that bandgap can close and reopen with switched topological winding numbers, as temperature increases, and demonstrate unambiguous signatures of topological edge states. These results connect entropic rigidity with topological band theory and establish a route to topological excitations that emerge upon heating.