Ergotropy of multipartite entangled states in the quantum atmosphere

We investigate the ergotropy of tripartite Greenberger-Horne-Zeilinger (GHZ) and W states in the quantum atmosphere of a Schwarzschild black hole, focusing on how the extractable work is redistributed under Hawking-induced thermalization. We find that the two classes of multipartite entanglement exhibit qualitatively different ergotropy structures. For GHZ states, the ergotropies associated with different accessible subsystems organize into two degenerate families, whereas for W states they obey a linear averaging relation with no counterpart in the GHZ case. Under symmetric gravitational thermal corrections, the two GHZ families remain separated by a constant energy gap that is independent of the radial position. We further show that the Hartle-Hawking constant D HH and the normalized radial distance r/r h play competing roles in the evolution of extractable work: increasing D HH suppresses its recovery toward the asymptotic Hawking limit, while increasing r/r h enhances the ergotropy accessible to the observers. These results show that the redistribution of extractable work in a black-hole environment is governed not only by gravitationally induced thermalization but also by the structure of the underlying multipartite entanglement, providing a thermodynamic signature that distinguishes GHZ- and W-type quantum correlations in curved spacetime.