Principally polarized abelian varieties of dimension $g$ are basic objects in algebraic geometry, but the cohomology of their moduli space $A_g$ is largely unknown. However, by a classical result of Borel, the cohomology of $A_g$ in degree $k < g$ is is freely generated by the odd Chern classes of the Hodge bundle. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of $A_g$, the Satake compactification. For most geometric applications, it is more natural to consider toroidal compactifications of $A_g$ instead. In this case, we have some stability results for the perfect cone compactification and the matroidal partial compactification. In this talk, we will consider the combinatorial aspects of this stable cohomology and its relationship with the structure of the toroidal fans.