  The proposed earlier relativistic mean-field model with hadron masses and
coupling constants depending on the $\sigma$-meson field is generalized to
finite temperatures. Within this approach we simulate the in-medium behavior of
the hadron masses motivated by the Brown-Rho scaling. The high-lying baryon
resonances and boson excitations as well as excitations of the $\sigma$,
$\omega$ and $\rho$ fields interacting via mean fields are incorporated into
this scheme. Thermodynamic properties of hot and dense hadronic matter are
elaborated with the constructed equation of state. Even at zero baryon density,
effective masses of $\sigma$-$\omega$-$\rho$-$N$ excitations abruptly drop down
for $T\gsim 170$ MeV and reach zero at a critical temperature
$T=T_{c\sigma}\sim 210$ MeV. Below $T_{c\sigma}$ (at $T\sim 190$ MeV) the
specific heat gets a peak like at crossover. We demonstrate that our EoS can be
matched with that computed on the lattice for high temperatures provided the
baryon resonance couplings with nucleon are partially suppressed. In this case
the quark liquid would masquerade as the hadron one. The model is applied to
description of heavy ion collisions in a broad collision energy range. It might
be especially helpful for studying phase diagram in the region near possible
phase transitions.
