Equipped with new powerful and efficient methods for computing quantum expectation values in staticspherically 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.