  In double perovskite oxides (A$_2$BB$^\prime$O$_6$), magnetism often arises
from diluted magnetic lattices, created by combining a perovskite structure
with localized 3$d$ magnetic elements (B) alongside another perovskite lattice
containing nearly nonmagnetic delocalized 4$d$/$5d$ elements (B$^\prime$).
Alternatively, the magnetic lattice can consist entirely of 3$d$ elements, with
one being completely nonmagnetic with $d^0$ state. La$_2$CoTiO$_6$ (LCTO), a
representative double perovskite oxide, contains Ti in a nonmagnetic state with
a $d^0$ electron configuration due to its $4^+$ oxidation state. Experimental
evidence shows that LCTO possesses a monoclinic structure (space group
$P2_1/n$) and behaves as an antiferromagnet with a N\'{e}el temperature of 14.6
K. Through first-principle electronic structure calculations, we uncover that
adjusting external hydrostatic pressure induces a sequence of phase
transitions: from antiferromagnetic insulator (AFM-I) to antiferromagnetic
metal (AFM-M), and ultimately to itinerant nonmagnetic metal (NM-M). The
transition from AFM-I to AFM-M at $\sim$ 42 GPa pressure coincides with a shift
in spin states, moving from a high-spin (HS) state to a low-spin (LS) state,
while Co retains a $d^7$ configuration. Distortion within the monoclinic
structure under pressure plays a pivotal role in the spin-state transition. At
the AFM-I to AFM-M transition, we observe a sharp decrease in the ratio of the
octahedral volumes occupied by Co and Ti. Such change in ratio is linked to
variations in octahedral volumes, akin to a breathing mode distortion. We
explore the impact of the breathing mode distortion by examining a highly
symmetric theoretical structure (space-group $I4/mmm$), achieved by optimizing
the structure with all $\angle${Co-O-Ti} angles set to 180$^{\circ}$.
