We prove that the topological entropy of the geodesic flow for a compact
Riemannian manifold (M, g) decreases as the metric g evolves under the
normalised Ricci flow provided that M admits a metric of constant negative
sectional curvature, and g is in a sufficiently small C^2 neighbourhood of the
constant curvature metric. More generally, the same phenomenon occurs if g
satisfies a certain negative curvature pinching condition, where the pinching
constant depends on both the dimension and the diameter of (M, g). This
provides an affirmative answer to an open question posed in Manning's paper
'The volume entropy of a surface decreases along the Ricci flow' [Ergodic
Theory Dynam. Systems, 24:171-76, 2004].