We show that for a smooth contact Anosov flow on a closed three manifold the measure of maximal entropy is in the Lebesgue class if and only if the flow is, up to finite covers, conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface. This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.