  We investigate the gravitational form factors (GFFs) of the nucleon and the
baryon octet, decomposed into their flavor components, utilizing a pion
mean-field approach grounded in the large $N_c$ limit of Quantum Chromodynamics
(QCD). Our focus is on the contributions from the twist-2 operators to the
flavor-triplet and octet GFFs, and we decompose the mass, angular momentum, and
$D$-term form factors of the nucleon into their respective flavors. The strange
quark contributions are found to be relatively mild for the mass and angular
momentum form factors, while providing significant corrections to the $D$-term
form factor. In the course of examining the flavor decomposition of the GFFs,
we uncover that the effects of twist-4 operators play a crucial role. While the
gluonic contributions are suppressed by the packing fraction of the instanton
vacuum in the twist-2 case, contributions from twist-4 operators are of order
unity, necessitating its explicit consideration.
