  Equipped with new powerful and efficient methods for computing quantum
expectation values in static-spherically symmetric spacetimes in arbitrary
dimensions, we perform an in-depth investigation of how the quantum vacuum
polarization varies with the parameters in the theory. In particular, we
compute and compare the vacuum polarization for a quantum scalar field in the
Schwarzschild anti-de Sitter black hole spacetime for a range of values of the
field mass and field coupling constant as well as the black hole mass and
number of spacetime dimensions. In addition, a new approximation for the vacuum
polarization in asymptotically anti-de Sitter black hole spacetimes is
presented.
