  The axion insulator is a higher-order topological insulator protected by
inversion symmetry. We show that under quenched disorder respecting inversion
symmetry {\it on average}, the topology of the axion insulator stays robust,
and an intermediate metallic phase in which states are delocalized is
unavoidable at the transition from an axion insulator to a trivial insulator.
We derive this conclusion from general arguments, from classical percolation
theory, and from the numerical study of a 3D quantum network model simulating a
disordered axion insulator through a layer construction. We find the
localization length critical exponent near the delocalization transition to be
$\nu=1.42\pm 0.12$. We further show that this delocalization transition is
stable even to weak breaking of the average inversion symmetry, up to a
critical strength. We also quantitatively map our quantum network model to an
effective Hamiltonian and we find its low energy k$\cdot$p expansion.
