  The electronic structure, phonons, electron-phonon coupling, and
superconductivity are theoretically studied in noncentrosymmetric
superconductor ThCoC$_2$ as a function of pressure, in the pressure range 0 -
20 GPa. We found that the electronic band splitting induced by the spin-orbit
coupling is enhanced under pressure. In spite of the overall stiffening of the
crystal lattice, the electron-phonon coupling constant $\lambda$ increases with
pressure, from 0.583 at 0 GPa to 0.652 at 20 GPa. If the isotropic Eliashberg
electron-phonon coupling theory is used to simulate the effect on the critical
temperature $T_c$, such an increase in $\lambda$ results in a substantial
increase of $T_c$ from 2.5~K at 0 GPa to 4~K at 20 GPa. This shows that
examining the effect of pressure offers a chance to resolve the pairing
mechanism in ThCoC$_2$.
