  We consider the codimension one asymptotic stability problem for the soliton
of the focusing cubic Klein-Gordon equation on the line under even
perturbations. The main obstruction to full asymptotic stability on the
center-stable manifold is a small divisor in a quadratic source term of the
perturbation equation. This singularity is due to the threshold resonance of
the linearized operator and the absence of null structure in the nonlinearity.
The threshold resonance of the linearized operator produces a one-dimensional
space of slowly decaying Klein-Gordon waves, relative to local norms. In
contrast, the closely related perturbation equation for the sine-Gordon kink
does exhibit null structure, which makes the corresponding quadratic source
term amenable to normal forms [76].
  The main result of this work establishes decay estimates up to exponential
time scales for small "codimension one type" perturbations of the soliton of
the focusing cubic Klein-Gordon equation. The proof is based upon a
super-symmetric approach to the study of modified scattering for 1D nonlinear
Klein-Gordon equations with P\"oschl-Teller potentials from [76], and an
implementation of a version of an adapted functional framework introduced in
[39].
