We study the thermodynamic properties of a frustrated Ising spin-5/2 hexagonal bipyramidal cluster. By employing exact enumeration, we investigate the ground-state phase diagram and the roles of thermal fluctuations and external magnetic field on the behavior of magnetization and entropy. The magnetocaloric effect of this nanosystem is also investigated by computing the isentropes in the temperature-magnetic field plane and the isothermal entropy change. The ground state exhibits magnetization plateaus at 1/4 and 1/2 of the saturation magnetization and a large degeneracy can be found at specific strengths of exchange couplings and magnetic field. The large ground-state degeneracy leads to an enhanced magnetocaloric effect manifested in the isentropic curves. More importantly, a significant reduction of temperature can be achieved by the isentropic removal of a very weak external field for a wide range of exchange couplings. Therefore, our results indicate that the frustrated spin-5/2 Ising hexagonal bipyramidal cluster is a nanostructure with potential application in efficient magnetic cooling.