  We study the anomalously large specific heat jump and its systematic change
with pressure in CeCoIn$_5$ superconductor. Starting with the general free
energy functional of the superconductor for a coupled electron boson system, we
derived the analytic result of the specific heat jump of the strong coupling
superconductivity occurring in the coupled electron boson system. Then using
the two component spin-fermion model we calculate the specific heat coefficient
$C(T)/T$ both for the normal and superconducting states and show a good
agreement with the experiment of CeCoIn$_5$. Our result also clearly
demonstrated that the specific heat coefficient $C(T)/T$ of a coupled electron
boson system can be freely interpreted as a renormalization either of the
electronic or of the bosonic degrees of freedom.
