Hyperelastic hinges or joints are of increasing interest for specific applications, where individual components can fulfill the function of an entire assembly. In individual cases, the design of such parts is a challenge and topology optimization can provide a first attempt for a more approachable design. Therefore, we apply thermodynamic topology optimization, an approach based on continuous density. Herein, the optimization is carried out by the solution of partial differential equations via a meshfree finite difference while the displacement-field is determined by classical finite element method (FEM). With regard to the function of the hinge, there are large deformations, especially in the context of a continuous density-based approach, and therefore snap-back and snap-through must be taken into account when solving the FEM. When solving the FEM, the Newton-Raphson algorithm including an arc-length method is used to approximate the non-linear problem, which results in multiple iteration steps within multiple load or time steps. Thus, the update for the design variables within the topology optimization can be applied at different times: within each Newton-Raphson increment or after each load step or only after the load step. The optimization in the final deformation, i.e., based on the last converged load step, is of course independent of the convergence speed or the number of load steps. The previous states of deformation are therefore not relevant in this case. Three approaches are applied for this purpose, which are based on the solution steps mentioned above: - optimization after each Newton-Raphson loop - optimization after each time step (with adapted number of increments) - optimization after the converged final time step These approaches are examined for varying (adaptive) time increments regarding their achieved optimally and computational performance. Numerical results are presented.