  Ap Librae is one out of a handful of low-frequency peaked blazars to be
detected at TeV $\gamma$-rays and the only one with an identified X-ray jet.
Combined observations of Fermi-LAT at high energies (HE) and of H.E.S.S at very
high energies (VHE) revealed a striking spectral property of Ap Librae; the
presence of a broad high-energy component that extends more than nine orders of
magnitude in energy and is, therefore, hard to be explained by the usual
single-zone synchrotron self-Compton model. We show that the superposition of
different emission components related to photohadronic interactions can explain
the $\gamma$-ray emission of Ap Librae without invoking external radiation
fields. We present two indicative model fits to the spectral energy
distribution of Ap Librae where the VHE emission is assumed to originate from a
compact, sub-pc scale region of the jet. A robust prediction of our model is
VHE flux variability on timescales similar to those observed at X-rays and HE
$\gamma$-rays, which can be further used to distinguish between a sub-pc or kpc
scale origin of the TeV emission. We thus calculate the expected variability
signatures at X-rays, HE and VHE $\gamma$-rays and show that quasi-simultaneous
flares are expected, with larger amplitude flares appearing at $\gamma$-rays.
We assess the detectability of VHE variability from Ap Librae with CTA, next
generation of IACTs. We show that $\sim$hr timescale variability at
$E_{\gamma}>0.1$ TeV could be detectable at high significance with shorter
exposure times than current Cherenkov telescopes.
