  CaSb$_2$ is the Dirac line-nodal material that exhibits a superconducting
(SC) transition at 1.7 K. In spite of its conventional SC state at ambient
pressure, the transition temperature $T_{\mathrm{c}}$ shows a peak structure
against hydrostatic pressure. We performed ac magnetic susceptibility and
$^{123}$Sb nuclear quadrupole resonance (NQR) measurements on
single-crystalline CaSb$_2$ under pressures up to 2.08 GPa. $T_{\mathrm{c}}$
monotonically increased in this pressure region, which is consistent with a
previous study. We observed continuous broadening of the NQR spectrum against
pressure, which is a sign of unique compression behavior of the lattice. In the
normal state, the nuclear spin-lattice relaxation rate 1/$T_1$ is proportional
to temperature in all pressure values; typical of a metal. However, 1/$T_1T$ in
the normal state is independent of pressure, indicating that the density of
states at the Fermi energy $N(E_{\mathrm{F}})$, which is one of the parameters
governing $T_{\mathrm{c}}$, is insensitive to pressure. From these results, we
conclude that $N(E_{\mathrm{F}})$ does not govern the origin of the enhancement
in $T_{\mathrm{c}}$. This is unusual for a weak electron-phonon coupling
superconductor. In the SC state, we revealed that the SC gap becomes larger and
more isotropic under pressure.
