  Bowen introduced a definition of topological entropy of subset inspired by
Hausdorff dimension in 1973 \cite{B}. In this paper we consider the Bowen's
entropy for amenable group action dynamical systems and show that under the
tempered condition, the Bowen entropy of the whole compact space for a given
F{\o}lner sequence equals to the topological entropy. For the proof of this
result, we establish a variational principle related to the Bowen entropy and
the Brin-Katok's local entropy formula for dynamical systems with amenable
group actions.
