  The thermodynamic properties of the bilayer Sr$_3$Ru$_2$O$_7$ at very low
temperatures are investigated by a realistic tight-binding model with the
on-site interactions treated at the mean-field level. Due to the strong
spin-orbit coupling, the band structure undergoes a significant change in Fermi
surface topology as the external magnetic field is applied, invalidating the
rigid band picture in which the Zeeman energy only causes chemical potential
shifts. In addition, since Sr$_3$Ru$_2$O$_7$ is a $t_{2g}$ active system with
unquenched orbital moments, the orbital Zeeman energy is not negligible and
plays an important role in the phase diagram on the magnetic field orientation.
We find that both the total density of states at the Fermi energy and the
entropy exhibit a sudden increase near the critical magnetic field for the
nematic phase, echoing the experimental findings. Our results suggest that
extra cares are necessary to isolate the contributions due to the quantum
criticality from the band structure singularity in this particular material.
The effects of quantum critical fluctuations are briefly discussed.
