  The Feynman-Metropolis-Teller treatment of compressed atoms has been recently
generalized to relativistic regimes and applied to the description of static
and rotating white dwarfs in general relativity. We present here the extension
of this treatment to the case of finite temperatures and construct the
corresponding equation of state (EOS) of the system; applicable in a wide
regime of densities that includes both white dwarfs and neutron star outer
crusts. We construct the mass-radius relation of white dwarfs at finite
temperatures obeying this new EOS and apply it to the analysis of ultra
low-mass white dwarfs with $M\lesssim 0.2 M_\odot$. In particular, we analyze
the case of the white dwarf companion of PSR J1738+0333. The formulation is
then extrapolated to compressed nuclear matter cores of stellar dimensions,
systems with mass numbers $A\approx (m_{\rm Planck}/m_n)^3$ or mass $M_{\rm
core}\approx M_{\odot}$, where $m_{\rm Planck}$ and $m_n$ are the Planck and
the nucleon mass. For $T \ll m_e c^2/k_B \approx 5.9\times 10^9$ K, a family of
equilibrium configurations can be obtained with analytic solutions of the
ultra-relativistic Thomas-Fermi equation at finite temperatures. Such
configurations fulfill global but not local charge neutrality and have strong
electric fields on the core surface. We find that the maximum electric field at
the core surface is enhanced at finite temperatures with respect to the
degenerate case.
