  We consider a quantum scalar field on the classical background of an
asymptotically anti-de Sitter black hole and the backreaction the field's
stress-energy tensor induces on the black hole geometry. The backreaction is
computed by solving the reduced-order semi-classical Einstein field equations
sourced by simple analytical approximations for the renormalized expectation
value of the scalar field stress-energy tensor. When the field is massless and
conformally coupled, we adopt Page's approximation to the renormalised stress
energy tensor while for massive fields, we adopt a modified version of the
DeWitt-Schwinger approximation. The latter approximation must be modified so
that it possesses the correct renormalization freedom required to ensure the
semi-classical equations are consistent. Equipped with these approximations,
the reduced-order field equations are easily integrated and the first-order (in
$\hbar$) corrections to the metric are obtained. We also compute the
corrections to the black hole event horizon, surface gravity and minimum
temperature as well as corrections to the photon sphere and quadratic curvature
invariants. We pay particular attention to the temperature profiles of the
semi-classical black holes compared with their classical counterparts, pointing
out some interesting qualitative features produced by the backreaction. These
results ought to provide reasonable approximations to the first-order
(one-loop) quantum backreaction on the geometry of asymptotically anti-de
Sitter black holes when the exact numerical stress-energy tensor sources the
semi-classical equations.
