Any surface, compact or otherwise, admits a metric with constant scalar curvature, so we can always find metrics with uniform density. Furthermore, any compact manifold admits a metric of constant scalar curvature in its conformal class, by the solution to the Yamabe problem.  On the other hand, the scalar curvature measures the deviation of the size of small geodesic balls of radius $\epsilon$ around a point compared to corresponding balls in Euclidean space,  but this deviation has size $\epsilon^{n+2}. As such, if the scalar curvature is non-constant, it's induced measure is non-uniform.