  We consider the stability of the toroidal AdS-Schwarzshild black holes as
solutions of the Einstein--Klein-Gordon system, with Dirichlet or Neumann
boundary conditions for the scalar field. Restricting to perturbations that
respect the toroidal symmetry we show both orbital and asymptotic stability for
the full nonlinear problem, for a range of choices of the Klein-Gordon mass.
The solutions we construct with Neumann boundary conditions have a Hawking mass
which diverges towards infinity, reflecting the infinite energy of the
Klein-Gordon field for perturbations satisfying these boundary conditions.
