  We show that Cooper pairing can occur intrinsically away from the Fermi
surface in $j=3/2$ superconductors with strong spin-orbit coupling and equally
curved bands in the normal state. In contrast to conventional pairing between
spin-$1/2$ electrons, we derive that pairing can happen between inter-band
electrons having different magnetic quantum numbers, for instance, $m_j=1/2$
and $m_j=3/2$. Such superconducting correlations manifest themselves by a pair
of indirect gap-like structures at finite excitation energies. An observable
signature of this exotic pairing is the emergence of a pair of symmetric
superconducting coherence peaks in the density of states at finite energies.
Moreover, the angular-momentum-resolved density of states in the presence of a
perturbative Zeeman field reflects the $m_j$ composition of the Cooper pairs.
We argue that such finite-energy pairing is a generic feature of $j=3/2$
superconductors, both in presence and absence of inversion symmetry.
