  As materials with suppressed ordering temperatures and enhanced ground state
entropies, frustrated magnetic oxides are ideal candidates for cryogenic
magnetocaloric refrigeration. While previous materials design has focused on
tuning the magnetic moments, their interactions, and density of moments on the
lattice, there has been relatively little attention to frustrated lattices.
Prior theoretical work has shown that the magnetocaloric cooling rate at the
saturation field is proportional to a macroscopic number of soft mode
excitations that arise due to the classical ground state degeneracy. The number
of these modes is directly determined by the geometry of the frustrating
lattice. For corner-sharing geometries, the pyrochlore has 50\% more modes than
the garnet and kagome lattices, whereas the edge-sharing \emph{fcc} has only a
subextensive number of soft modes. Here, we study the role of soft modes in the
magnetocaloric effect of four large-spin Gd$^{3+}$ ($L=0$, $J=S=7/2$)
Heisenberg antiferromagnets on a kagome, garnet, pyrochlore, and \emph{fcc}
lattice. By comparing measurements of the magnetic entropy change $\Delta S_m$
of these materials at fields up to $9$~T with predictions using mean-field
theory and Monte Carlo simulations, we are able to understand the relative
importance of spin correlations and quantization effects. We observe that
tuning the value of the nearest neighbor coupling has a more dominant
contribution to the magnetocaloric entropy change in the liquid-He cooling
regime ($2$-$20$~K), rather than tuning the number of soft mode excitations.
Our results inform future materials design in terms of dimensionality, degree
of magnetic frustration, and lattice geometry.
