  The resource theory with covariant Gibbs-preserving operations, also called
enhanced thermal operations, is investigated. We prove that with the help of a
correlated catalyst, the state convertibility for any coherent state is fully
characterized by the free energy defined with the quantum relative entropy. We
can extend this result to general resource theories in the form that imposing
the covariant condition to a general resource theory does not change the state
convertibility as long as the initial state is coherent and distillable. This
means that the additional constraint from the law of energy conservation is
irrelevant in the correlated-catalytic framework.
