  Given a closed, orientable surface of constant negative curvature and genus
$g \ge 2$, we study a family of generalized Bowen-Series boundary maps and
prove the following rigidity result: in this family the topological entropy is
constant and depends only on the genus of the surface. We give an explicit
formula for this entropy and show that the value of the topological entropy
also stays constant in the Teichm\"uller space of the surface. The proofs use
conjugation to maps of constant slope.
