  Superconductors without inversion symmetry in their crystal structure are
known to exhibit unconventional properties. Recently, based on the measured
temperature dependence of the magnetic field penetration depth,
superconductivity in noncentrosymmetric $\mathrm{ThCoC_2}$ was proposed to be a
nodal $d$-wave and mediated by the spin fluctuations. Moreover, a non-BCS
behavior of the temperature dependence of the electronic specific heat and the
magnetic upper critical field were reported. In this work, the electronic
structure, phonons and electron-phonon coupling are studied in
$\mathrm{ThCoC_2}$ on the basis of \textit{ab initio} computations. The effect
of the spin-orbit coupling on the electronic structure and electron-phonon
interaction is analyzed, and a large splitting of the electronic band structure
is found. The calculated electron-phonon coupling constant $\lambda = 0.59$
remains in decent agreement with the experimental estimates, suggesting that
the electron-phonon interaction is strong enough to explain superconductivity
with $T_c \simeq 2.5$~K. Nevertheless, we show that the conventional isotropic
Eliashberg formalism is unable to describe the thermodynamic properties of the
superconducting state, as calculated temperature dependence of the electronic
specific heat and magnetic penetration depth deviate from experiments, which is
likely driven by the strong spin-orbit coupling and inversion symmetry
breaking. In addition, to shed more light on the pairing mechanism, we propose
to measure the carbon isotope effect, as our calculations based on the
electron-phonon coupling predict the observation of the isotope effect with an
exponent $\alpha \simeq 0.15$.
