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I will discuss the cosmological (de)confinement phase transition (PT) of nearly conformal, strongly coupled large N field theories, applicable to composite Higgs models, using their holographic dual 5D formulation. In this description, the PT is from the high temperature phase in which the IR of the warped extra dimension is covered by a black-brane horizon to the low temperature Randall-Sundrum-1 phase. The PT proceeds by percolation of IR-brane bubbles nucleating from the horizon, and the bubble dynamics during the PT sources a stochastic gravitational wave background that can be detected by future experiments. I will show how to construct a smooth “bounce” configuration that interpolates between the two phases, allowing for a controlled description of bubble nucleation within 5D EFT. We will see that the cosmological PT in the minimal models can complete only after a large period of supercooling, potentially resulting in excessive dilution of primordial matter abundances. I will then show how generic modifications of the minimal models can result in a much faster completion of the PT. Finally, I will discuss the gravitational wave signal produced by the PT and the implications of the PT for baryogenesis.