We study the link between variations of entropy on a compact hyperbolic manifold $M$ and infinitesimal flat conformal deformations of $M$. We remark that the entropy of the Liouville measure increases in the direction of these deformations. We then explain a new construction of infinitesimal flat conformal deformations by bending along totally geodesic hypersurfaces of $M$ which allows us to extend a theorem of L. Flaminio.