  We report wide-doping-range ($8 \times 10^{17}$ to $4 \times 10^{20}$
cm$^{-3}$ Hall electron density) low temperature specific heat measurements on
single crystal SrTiO$_3$:Nb, correlated with electronic transport data and
tight-binding modeling. Lattice dynamic contributions to specific heat are
shown to be well understood, albeit with unusual sensitivity to doping, likely
related to the behavior of soft modes. Electronic contributions to specific
heat provide effective masses that increase substantially, from $1.8$ to $4.8
m_e$, across the two SrTiO$_3$ Lifshitz transitions. It is shown that this
behavior can be quantitatively reconciled with quantum oscillation data and
calculated band structure, establishing a remarkably doping-independent mass
enhancement factor of $2.0$. Most importantly, with the doping-dependent $T^2$
resistivity prefactor and Sommerfeld coefficient known, Kadowaki-Woods scaling
has been tested over the entire doping range probed. Despite classic Fermi
liquid behavior in electronic specific heat, standard Kadowaki-Woods scaling is
dramatically violated, highlighting the need for new theoretical descriptions
of $T^2$ resistivity in SrTiO$_3$.
