V-static metrics and the volume-renormalised mass

V-static metrics generalise the notion of static metrics, and stem from the work of Miao and Tam [arXiv:0807.2693], and Corvino, Eichmair and Miao [arXiv:1211.6168] on critical points of the volume functional over the space of compact manifolds with constant scalar curvature. In this article we show that these V-static metrics arise naturally in the context of asymptotically hyperbolic manifolds as critical points of the volume-renormalised mass, recently introduced by Dahl, Kröncke and the author [arXiv:2307.06196]. In particular, we show that critical points of the volume-renormalised mass over the space of constant scalar curvature asymptotically hyperbolic manifolds without boundary, or satisfying appropriate boundary conditions, are exactly V-static metrics. This is directly analogous to the relationship between critical points of the ADM mass and static metrics for asymptotically flat manifolds.