The radio lobes of Hydra A lie within cavities surrounded by a rim of enhanced X-ray emission in the intracluster gas. Although the bright rim appears cooler than the surrounding gas, existing Chandra data do not exclude the possibility that the rim is produced by a weak shock. A temperature map shows that cool gas extends out along the radio axis of Hydra A. The age of the radio source and equipartition pressure of the radio lobe argue against a shock, and comparison with similar structure in the Perseus Cluster also suggests that the rim is cool. We show that the cool bright rim cannot be the result of shock-induced cooling or due to the effect of magnetic fields in shocks. The most likely source of low-entropy (cool) gas is entrainment by the rising cavity. This requires some means of communicating the bouyant force on the cavity to the surrounding gas. The magnetic field required to produce the Faraday rotation in Hydra A has the appropriate properties for this if the Faraday screen is mainly in this bright rim. In Hydra A, the mass outflow due to the rising cavities could be sufficient to balance cooling-driven inflow, so preventing the build up of low-entropy gas in the cluster core.