  For a many-body system of arbitrary dimension, we consider fermionic ground
states of non-interacting Hamiltonians invariant under a $C_2$ cyclic group.
The absolute difference $\Delta$ between the number of occupied symmetric and
anti-symmetric single-particle states is an adiabatic invariant. We prove lower
bounds on the configurational and the number entropy based on this invariant.
In band insulators, the topological invariant $\Delta$ and the entropy bounds
can be directly determined from high symmetry points in the Brillouin zone.
