  In this paper, we report the measurements of specific heat of an amorphous
$Ti_{9.5}Si_{90.5}$ alloy located very close to the critical point of the
metal-insulator transition. In the presence of a magnetic field, the specific
heat is dominated by the Schottky anomaly caused by magnetic moments associated
with the dangling bonds in the matrix of amorphous Si. Subtraction of this
contribution exposes the behavior of the electronic specific heat coefficient
$\gamma$. The coefficient is temperature-independent above 2 K and is, in order
of magnitude, close to the value expected in the absence of electron-electron
interactions. In the temperature range 0.4-1.5 K, the coefficient $\gamma$
shows an anomalous downturn, which can be approximated by the dependence
$\gamma\left(T\right)=\gamma_0ln{(T/T_0)}$, with $T_0\approx0.2$ K . In a
companion paper, we found that the Hall coefficient in Ti-Si alloys is affected
by the electron-electron interaction up to much higher temperature of 150 K and
also varies critically across the metal-insulator transition. We compare our
results with theoretical predictions for three models, which can potentially
explain the anomalous behavior of the specific heat: generalized non-linear
$\sigma$ model, Coulomb glass, and many-body localization.
