  On $\text{AdS}_2 \times \mathbb{S}^2$, we construct the two-point correlation
functions for the ground and thermal states of a real Klein-Gordon field
admitting generalized $(\gamma,v)$-boundary conditions. We follow the
prescription recently outlined in [1] for two different choices of secondary
solutions. For each of them, we obtain a family of admissible boundary
conditions parametrized by $\gamma\in\left[0,\frac{\pi}{2}\right]$. We study
how they affect the response of a static Unruh-DeWitt detector. The latter not
only perceives variations of $\gamma$, but also distinguishes between the two
families of secondary solutions in a qualitatively different, and rather
bizarre, fashion. Our results highlight once more the existence of a freedom in
choosing boundary conditions at a timelike boundary which is greater than
expected and with a notable associated physical significance.
