  Recent work has shown that certain deformations of the scalar potential in
Jackiw-Teitelboim gravity can be written as double-scaled matrix models.
However, some of the deformations exhibit an apparent breakdown of unitarity in
the form of a negative spectral density at disc order. We show here that the
source of the problem is the presence of a multi-valued solution of the leading
order matrix model string equation. While for a class of deformations we fix
the problem by identifying a first order phase transition, for others we show
that the theory is both perturbatively and non-perturbatively inconsistent.
Aspects of the phase structure of the deformations are mapped out, using
methods known to supply a non-perturbative definition of undeformed JT gravity.
Some features are in qualitative agreement with a semi-classical analysis of
the phase structure of two-dimensional black holes in these deformed theories.
