  We present a new mode-sum regularization prescription for computing the
vacuum polarization of a scalar field in static spherically-symmetric black
hole spacetimes in odd dimensions. This is the first general and systematic
approach to regularized vacuum polarization in higher dimensions. Remarkably,
the regularization parameters can be computed in closed form in arbitrary
dimensions and for arbitrary metric function $f(r)$. In fact, we show that in
spite of the increasing severity and number of the divergences to be
regularized, the method presented is mostly agnostic to the number of
dimensions. Finally, as an explicit example of our method, we show plots for
vacuum polarization in the Schwarzschild-Tangherlini spacetime for odd
$d=5,...,11$.
