Dynamic capacity allocation control for resource sharing networks (RSN) is studied when the networks are in heavy traffic. The goal is to minimize an ergodic cost with a linear holding cost function. Our main result shows that the optimal cost associated with an associated Brownian control problem provides a lower bound for the asymptotic ergodic cost in the RSN for any sequence of control policies. A similar result for an infinite horizon discounted cost has been previously shown for general resource sharing networks and for general "unitary networks." The study of an ergodic cost criterion requires different ideas as bounds on ergodic costs only yield an estimate on the controls in a time-averaged sense which makes the time-rescaling ideas used in the infinite horizon discounted cost cases hard to implement. Proofs rely on working with a weaker topology on the space of controlled processes that is more amenable to an analysis for the ergodic cost. As a corollary of the main result, we show that control policies which achieve hierarchical greedy ideal performance for general RSN are asymptotically optimal for the ergodic cost when the underlying cost per unit time has certain monotonicity properties.