  We report a study on the specific heat and heat transport of
Tb$_2$Ti$_{2-x}$Zr$_x$O$_7$ ($x =$ 0, 0.02, 0.1, 0.2, and 0.4) single crystals
at low temperatures and in high magnetic fields. The magnetic specific heat can
be described by the Schottky contribution from the crystal-electric-field (CEF)
levels of Tb$^{3+}$, with introducing Gaussian distributions of the energy
split of the ground-state doublet and the gap between the ground state and
first excited level. These crystals has an extremely low phonon thermal
conductivity in a broad temperature range that can be attributed to the
scattering by the magnetic excitations, which are mainly associated with the
CEF levels. There is strong magnetic field dependence of thermal conductivity,
which is more likely related to the field-induced changes of phonon scattering
by the CEF levels than magnetic transitions or spin excitations. For magnetic
field along the [111] direction, there is large thermal Hall conductivity at
low temperatures which displays a broad peak around 8 T. At high fields up to
14 T, the thermal Hall conductivity decreases to zero, which supports its
origin from either the spinon transport or the phonon skew scattering by CEF
levels. The thermal Hall effect is rather robust with Zr doping up to 0.2 but
is strongly weakened in higher Zr-doped sample.
