We consider the Klein-Gordon equation on the exterior of the toric anti
de-Sitter Schwarzschild black hole with Dirichlet, Neumann and Robin boundary conditions
at I. We define a non-degenerate energy for the equation which controls the
renormalised H1 norm of the field. We then establish both decay and integrated decay
of this energy through vector field methods. Finally we demonstrate the necessity
of ‘losing a derivative’ in the integrated energy estimate through the construction of
a Gaussian beam staying in the exterior of the event horizon for arbitrary long coordinate
time.