  We investigate grain boundaries (GBs) in the cubic inverse Heusler phase
Fe$_2$CoGa by means of first principles calculations based on density
functional theory. Besides the energetic stability, the analysis focuses on the
magnetic properties of a set of 16 GB structures in this intermetallic phase.
We determine the integrated excess magnetization across the GB and analyze it
in terms of the projected local magnetic moments of the atoms and their local
Voronoi volumes. The results are systematically compared to those of
corresponding GBs in body-centered cubic (bcc) Fe. The studied GBs in
Fe$_2$CoGa may have a considerably increased magnetization at the GB, up to
more than twice as much as in bcc Fe, depending on the GB type, while
geometrical quantities like GB widening or local GB excess volume distributions
are similar for both phases. We explain this difference by the higher
flexibility of the ternary Fe$_2$CoGa phase in compensating the disturbance of
a crystal defect by structural relaxation. The GB structures therefore have a
lower energy accompanied with increased local magnetic moments of the Co and
half of the Fe atoms within a distance of a few angstrom around the GB plane.
