It is by now a classical topic in index theory to study on a (closed) Riemannian (spin) manifold the space of all Riemannian metrics of positive scalar curvature. We have several results showing that this space is usually highly complicated from a homotopy theoretic point of view (provided it is non-empty).
Instead of studying positivity of scalar curvature, one can of course also study positivity of Ricci curvature or positivity of sectional curvature. But here our results about the topology of these spaces are scarce.
In a recent preprint on the arXiv (2007.15062) Krannich, Kupers and Randal-Williams manage to show that the space of Riemannian metrics with positive sectional curvature (resp. positive Ricci curvature) can have non-trivial rational higher homotopy groups.
update (October 12th, 2020): a similar, related result of Frenck-Reinhold may be found here: arXiv:2010.04588.