  Aubry-Andre Harper (AAH) lattice models, characterized by
reflection-asymmetric, sinusoidally varying nearest-neighbor tunneling profile,
are well-known for their topological properties. We consider the fate of such
models in the presence of balanced gain and loss potentials $\pm i\gamma$
located at reflection-symmetric sites. We predict that these models have a
finite $\mathcal{PT}$ breaking threshold only for {\it specific locations} of
the gain-loss potential, and uncover a hidden symmetry that is instrumental to
the finite threshold strength. We also show that the topological edge-states
remain robust in the $\mathcal{PT}$-symmetry broken phase. Our predictions
substantially broaden the possible realizations of a $\mathcal{PT}$-symmetric
system.
