  By refining the method proposed in arXiv:2010.07660, entropy current and
entropy density for a relativistic hydrostatic equilibrium system with
spherical symmetry are constructed as a non-Noether conserved charge in the
Einstein gravity with cosmological constant. It is shown that the constructed
entropy density satisfies both the local Euler relation and the first law of
thermodynamics non-perturbatively with respect to the Newton constant. Finally
the established relativistic thermodynamics is applied to one of the systems
with uniform energy density as a crude model of a degenerate star and its local
thermodynamic observables are determined analytically.
