We propose dual thermodynamics corresponding to black hole mechanics with the identifications E′ → A/4, S′ → M, and T′ → T-1 in Planck units. Here A, M and T are the horizon area, mass and Hawking temperature of a black hole and E′, S′ and T′ are the energy, entropy and temperature of a corresponding dual quantum system. We show that, for a Schwarzschild black hole, the dual variables formally satisfy all three laws of thermodynamics, including the Planck-Nernst form of the third law requiring that the entropy tend to zero at low temperature. Once the third law is satisfied, it is straightforward to construct simple (dual) quantum systems representing black hole mechanics. In addition to recovering black hole mechanics, we obtain quantum corrections to the entropy, including the logarithmic correction obtained by previous authors.