  We study static, spherically symmetric black hole solutions of the Einstein
equations with a positive cosmological constant and a conformally coupled self
interacting scalar field. Exact solutions for this model found by
Mart{\'\i}nez, Troncoso, and Zanelli, (MTZ), were subsequently shown to be
unstable under linear perturbations, with modes that diverge arbitrarily fast.
We find that the moduli space of static, spherically symmetric solutions that
have a regular horizon -and satisfy the weak and dominant energy conditions
outside the horizon- is a singular subset of a two dimensional space
parameterized by the horizon radius and the value of the scalar field at the
horizon. The singularity of this space of solutions provides an explanation for
the instability of the MTZ spacetimes, and leads to the conclusion that, if we
include stability as a criterion, there are no physically acceptable black hole
solutions for this system that contain a cosmological horizon in the exterior
of its event horizon.
