Given a closed, orientable surface of constant negative curvature and genus g ≥ 2, we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to their smooth invariant measure. Each such map is defined for a particular fundamental polygon  and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy  is constant in this family and depends only on the genus of the surface. We give an explicit formula for this entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy for the smooth invariant probability measure on the circle that only depends on the genus of the surface and the perimeter of the (8g−4)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental (8g-4)-gon, and stays constant  on a subset of the parameter space. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof - valid only for certain multi-parameters - uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map, and the Reduction Theory for maps associated to Fuchsian groups.