  We develop a general approach to compute the symmetry-resolved R\'enyi and
von Neumann entanglement entropies (SREE) of thermodynamic macrostates in
interacting integrable systems. Our method is based on a combination of the
thermodynamic Bethe ansatz and the G\"artner-Ellis theorem from large deviation
theory. We derive an explicit simple formula for the von Neumann SREE, which we
show to coincide with the thermodynamic Yang-Yang entropy of an effective
macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin
chain, we test our result against iTEBD calculations for thermal states,
finding good agreement. As an application, we provide analytic predictions for
the asymptotic value of the SREE following a quantum quench.
