  Given a closed, oriented, compact surface $S$ of constant negative curvature
and genus $g \ge 2$, we study the measure-theoretic entropy of the Bowen-Series
boundary map with respect to its smooth invariant measure. We obtain an
explicit formula for the entropy that only depends on the perimeter of the
$(8g-4)$-sided fundamental polygon of the surface $S$ and its genus. Using
this, we analyze how the entropy changes in the Teichm\"uller space of $S$ and
prove the following flexibility result: the measure-theoretic entropy takes all
values between $0$ and a maximum that is achieved on the surface that admits a
regular $(8g-4)$-sided fundamental polygon. We also compare the
measure-theoretic entropy to the topological entropy of these maps and show
that the smooth invariant measure is not the measure of maximal entropy.
