  We examine uniqueness of equilibrium states for the natural extension of a
topologically exact, non-uniformly expanding, local homeomorphism with a
H\"older continuous potential function. We do this by applying general
techniques developed by Climenhaga and Thompson, and show there is a natural
condition on decompositions that guarantees that a unique equilibrium state
exists. We then show how to apply these results to partially hyperbolic
attractors.
