  Packing topological entropy is a dynamical analogy of the packing dimension,
which can be viewed as a counterpart of Bowen topological entropy. In the
present paper, we will give a systematically study to the packing topological
entropy for a continuous $G$-action dynamical system $(X,G)$, where $X$ is a
compact metric space and $G$ is a countable discrete amenable group.
  We first prove a variational principle for amenable packing topological
entropy: for any Borel subset $Z$ of $X$, the packing topological entropy of
$Z$ equals the supremum of upper local entropy over all Borel probability
measures for which the subset $Z$ has full measure. And then we obtain an
entropy inequality concerning amenable packing entropy. Finally we show that
the packing topological entropy of the set of generic points for any invariant
Borel probability measure $\mu$ coincides with the metric entropy if either
$\mu$ is ergodic or the system satisfies a kind of specification property.
