  In this paper, we establish a variational principle, between the fiber
Bowen's topological entropy on conditional level sets of Birkhoff average and
fiber measure-theoretical entropy, for the skew product transformation driven
by a uniquely ergodic homeomorphism system satisfying Anosov and topological
mixing on fibers property. We prove it by utilizing a fiber specification
property. Moreover, we prove that such skew product transformation has
specification property defined by Gundlach and Kifer. Employing their main
results, every H\"older continuous potential has a unique equilibrium state,
and we also establish a variational principle between the fiber
measure-theoretic entropy and the fiber Bowen's topological entropy on
conditional level sets of local entropy for such unique equilibrium state.
Examples of systems under consideration are given, such as fiber Anosov maps on
2-dimension tori driven by any irrational rotation on circle and random
composition of 2x2 area preserving positive matrices driven by uniquely ergodic
subshift.
