  Thermodynamic characteristics of Fermi systems are investigated in the
vicinity of a phase transition where the effective mass diverges and the
single-particle spectrum becomes flat. It is demonstrated that at extremely low
temperatures $T$, the flattening of the spectrum is reflected in
non-Fermi-liquid behavior of the inverse susceptibility $\chi^{-1}(T) \sim
T^{\alpha}$ and the specific heat $C(T)/T\sim T^{-\alpha}$, with the critical
index $\alpha=2/3$. In the presence of an external static magnetic field $H$,
both these quantities are found to exhibit a scaling behavior, e.g.
$\chi^{-1}(T,H)=\chi^{-1}(T,0)+ T^{2/3}F(H/T)$, in agreement with available
experimental data.
