  We show that the Einstein-Hilbert functional, as a functional on the space of
Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In
particular, this provides an obstruction to the existence of a constant scalar
curvature Sasakian metric. As an application we prove that K-semistable
polarized Sasaki manifold has vanishing Sasaki-Futaki invariant. We then apply
this result to show that under the right conditions on the Sasaki join
manifolds of [7] a polarized Sasaki manifold is K-semistable if only if it has
constant scalar curvature.
