  We study the entanglement spectrum (ES) of two-dimensional $C_{n}$-symmetric
second-order topological insulators (TIs). We show that some characteristic
higher order topological observables, e.g., the filling anomaly and its
associated fractional corner charge, can be determined from the ES of atomic
and fragile TIs. By constructing the relationship between the configuration of
Wannier orbitals and the number of protected in-gap states in the ES for
different symmetric cuts in real space, we express the fractional corner charge
in terms of the number of protected in-gap states of the ES. We show that our
formula is robust in the presence of electron-electron interactions as long as
the interactions preserve $C_{n}$ rotation symmetry and charge-conservation
symmetry. Moreover, we discuss the possible signatures higher order topology in
the many-body ES. Our methods allow the identification of some classes of
higher order topology without requiring the usage of nested Wilson loops or
nested entanglement spectra.
