In the reversible Oregonator, a model exhibiting oscillations in the Belousov-Zhabotinskii reaction, a bistability in the entropy production (free energy dissipation) between a steady state and an oscillatory state is reported in the neighborhood of a subcritical Hopf bifurcation. We identified this region in the past by a perturbation method. The numerical results present a thermodynamic formulation of multistable structures in a nonlinear dissipative system.