  Although numerical studies modeling the quantum hall effect at filling
fraction $5/2$ predict either the Pfaffian (Pf) or its particle hole conjugate,
the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantized
thermal hall conductivity with quantum number $K=5/2$ , rather than the value
$K=7/2$ or $K=3/2$ expected for the Pf or aPF state, respectively. While a
particle hole symmetric topological order (the PH-Pfaffian) would be consistent
with the experiments, this state is believed to be energetically unfavorable in
a homogenous system. Here we study the effects of disorder that are assumed to
locally nucleate domains of Pf and aPf. When the disorder is relatively weak
and the size of domains is relatively large, we find that when the electrical
Hall conductance is on the quantized plateau with $\sigma_{xy} = (5/2)(e^2/h)$,
the value of $K$ can be only 7/2 or 3/2, with a possible first-order-like
transition between them as the magnetic field is varied. However, for
sufficiently strong disorder an intermediate state might appear, which we
analyze within a network model of the domain walls. Predominantly, we find a
thermal metal phase, where $K$ varies continuously and the longitudinal thermal
conductivity is non-zero, while the electrical Hall conductivity remains
quantized at $(5/2)e^2/h$. However, in a restricted parameter range we find a
thermal insulator with $K=5/2$, a disorder stabilized phase which is
adiabatically connected to the PH-Pfaffian. We discuss a possible scenario to
rationalize these special values of parameters.
