We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds
$(M,g)$
when the conformal boundary
$\unicode[STIX]{x2202}M$
has dimension
$n$
even. Its definition depends on the choice of metric
$h_{0}$
on
$\unicode[STIX]{x2202}M$
in the conformal class at infinity determined by
$g$
, we denote it by
$\text{Vol}_{R}(M,g;h_{0})$
. We show that
$\text{Vol}_{R}(M,g;\cdot )$
is a functional admitting a ‘Polyakov type’ formula in the conformal class
$[h_{0}]$
and we describe the critical points as solutions of some non-linear equation
$v_{n}(h_{0})=\text{constant}$
, satisfied in particular by Einstein metrics. When
$n=2$
, choosing extremizers in the conformal class amounts to uniformizing the surface, while if
$n=4$
this amounts to solving the
$\unicode[STIX]{x1D70E}_{2}$
-Yamabe problem. Next, we consider the variation of
$\text{Vol}_{R}(M,\cdot ;\cdot )$
along a curve of AHE metrics
$g^{t}$
with boundary metric
$h_{0}^{t}$
and we use this to show that, provided conformal classes can be (locally) parametrized by metrics
$h$
solving
$v_{n}(h)=\text{constant}$
and
$\text{Vol}(\unicode[STIX]{x2202}M,h)=1$
, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space
${\mathcal{T}}(\unicode[STIX]{x2202}M)$
of conformal structures on
$\unicode[STIX]{x2202}M$
. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.