  The origin of the partial suppression of the electronic density states in the
enigmatic pseudogap behavior, which is at the core of understanding high-$T_c$
superconductivity, has been hotly contested as either a hallmark of preformed
Cooper pairs or an incipient order of competing interactions nearby. Here, we
report the quasi-particle scattering spectroscopy of the quantum critical
superconductor CeCoIn$_5$, where a pseudogap with energy $\Delta_g$ was
manifested as a dip in the differential conductance ($dI/dV$) below the
characteristic temperature of $T_g$. When subjected to external pressure, $T_g$
and $\Delta_g$ gradually increase, following the trend of increase in quantum
entangled hybridization between Ce 4$f$ moment and conduction electrons. On the
other hand, the superconducting (SC) energy gap and its phase transition
temperature shows a maximum, revealing a dome shape under pressure. The
disparate dependence on pressure between the two quantum states shows that the
pseudogap is less likely involved in the formation of SC Cooper pairs, but
rather is controlled by Kondo hybridization, indicating that a novel type of
pseudogap is realized in CeCoIn$_5$.
