  We present a rigidity theorem for the action of the mapping class group
$\pi_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of
positive scalar curvature for high dimensional manifolds $M$. This result is
applicable to a great number of cases, for example to simply connected
$6$-manifolds and high dimensional spheres. Our proof is fairly direct, using
results from parametrised Morse theory, the $2$-index theorem and computations
on certain metrics on the sphere. We also give a non-triviality criterion and a
classification of the action for simply connected $7$-dimensional
$\mathrm{Spin}$-manifolds.
