Summary: The Riemannian barycentre (or Fréchet mean) is the workhorse of data analysis for data taking values in Riemannian manifolds. The Riemannian barycentre of a probability distribution \(P\) on a Riemannian manifold \(M\) is a possible generalisation of the concept of expected value, at least when the barycentre is unique. Knowing when the barycentre of \(P\) is unique is of fundamental importance for its interpretation and computation. Existing results can only guarantee this uniqueness by assuming \(P\) is supported inside a convex geodesic ball \(B(x^*, \delta) \subset M\). This assumption is overly restrictive since many distributions have support equal to \(M\) yet are sufficiently concentrated within a convex geodesic ball that they nevertheless have a unique barycentre. This paper studies the concentration of Gibbs distributions on Riemannian manifolds and gives conditions for the barycentre to be unique. Specifically, consider the Gibbs distribution \(P =P_T\) with unnormalised density \(\exp(-U/T)\) for some potential \(U:M\rightarrow\mathbb{R}\) and some temperature \(T>0\). If \(M\) is a simply connected compact Riemannian symmetric space, and \(U\) has a unique global minimum at \(x^*\), then for each \(\delta<\frac{1}{2}r_{cx}\) (\(r_{cx}\) the convexity radius of \(M\)), there exists a critical temperature \(T_\delta\) such that \(T<T_\delta\) implies \(P_T\) has a unique Riemannian barycentre \(\bar{x}_T\) and this \(\bar{x}_T\) belongs to the geodesic ball \(B(x^*, \delta)\). Moreover, if \(U\) is invariant by geodesic symmetry about \(x^*\), then \(\bar{x}_T = x^*\). Remarkably, this conclusion does not require the potential \(U\) to be smooth and therefore serves as the foundation of a new general algorithm for black-box optimisation. This algorithm is briefly illustrated with two numerical experiments.