  Given an integer $k \geq 5$, and a $C^k$ Anosov flow $\Phi$ on some compact
connected $3$-manifold preserving a smooth volume, we show that the measure of
maximal entropy (MME) is the volume measure if and only if $\Phi$ is
$C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$
arbitrarily small. Besides the rigidity, we also study the entropy flexibility,
and show that the metric entropy with respect to the volume measure and the
topological entropy of suspension flows over Anosov diffeomorphisms on the
$2$-torus achieve all possible values subject to natural normalizations.
Moreover, in the case of dispersing billiards, we show that if the measure of
maximal entropy is the volume measure, then the Birkhoff Normal Form of regular
periodic orbits with a homoclinic intersection is linear.
