A topological superconductor candidate $beta$-RhPb$_2$ is predicted by using the first-principles electronic structure calculations. Our calculations show that there is a band inversion around the Fermi level at the Z point of Brillouin zone. The calculated nonzero topological invariant Z$_2$ indicates that $beta$-RhPb$_2$ is a topological insulator defined on a curved Fermi level. The slab calculations further demonstrate that the gapless nontrivial topological surface states (TSS) are not overlapped by the bulk states and they cross the Fermi level. The phonon calculations confirm the dynamical stability of $beta$-RhPb$_2$, while the electron-phonon coupling (EPC) calculations predict that the superconducting transition temperature ($T_c$) of $beta$-RhPb$_2$ can reach 9.7 K. The coexistence of nontrivial topological band structure with the TSS crossing the Fermi level as well as the superconducting $T_c$ above the liquid-helium temperature suggest that the layered compound $beta$-RhPb$_2$ is a topological superconductor, which deserves further experimental verification.
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