  We continue the study in a previous work in the setting of weighted
pluripotential theory arising from polynomials associated to a convex body $P$
in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in
this setting specifying the rate function in terms of
$P-$pluripotential-theoretic notions. As an important preliminary step, we
first give an existence proof for the solution of a Monge-Amp\`ere equation in
an appropriate finite energy class. This is achieved using a variational
approach.
