  We identify a class of Randall-Sundrum type models with a successful first
order cosmological phase transition during which a 5D dual of approximate
conformal symmetry is spontaneously broken. Our focus is on soft-wall models
that naturally realize a light radion/dilaton and suppressed dynamical
contribution to the cosmological constant. We discuss phenomenology of the
phase transition after developing a theoretical and numerical analysis of these
models both at zero and finite temperature. We demonstrate a model with a
TeV-Planck hierarchy and with a successful cosmological phase transition where
the UV value of the curvature corresponds, via AdS/CFT, to an $N$ of $20$,
where 5D gravity is expected to be firmly in the perturbative regime.
