  A self-consistent description for hot $\Lambda$ hypernuclei in hypothetical
big boxes is developed within the relativistic Thomas-Fermi approximation in
order to investigate directly the liquid-gas phase coexistence in strangeness
finite nuclear systems. We use the relativistic mean-field model for nuclear
interactions. The temperature dependence of $\Lambda$ hyperon density,
$\Lambda$ hyperon radius, excitation energies, specific heat, and the binding
energies of $\Lambda$ hypernuclei from $^{16}_{\Lambda}$O to
$^{208}_{\Lambda}$Pb in phase transition region are calculated by using the
subtraction procedure in order to separate the hypernucleus from the
surrounding baryon gas. The $\Lambda$ central density is very sensitive to the
temperature. The radii of $\Lambda$ hyperon at high temperature become very
large. In the relativistic Thomas-Fermi approximation with the subtraction
procedure, the properties of hypernuclei are independent of the size of the box
in which the calculation is performed. The level density parameters of
hypernuclei in the present work are confirmed to be almost constant at low
temperature. It is also found that the single-$\Lambda$ binding energies of
$\Lambda$ hypernuclei are largely reduced with increasing temperature.
