  In this paper we study phase transitions in a maximally symmetric composite
Higgs model with next-to-minimal coset, where a pseudoscalar singlet emerges
alongside the Higgs doublet. The maximal symmetry guarantees the finiteness of
the radiatively generated scalar potential. We explore the scenario involving
an explicit source of CP violation in the strong sector, which induces a
$\mathbb{Z}_2$ asymmetric scalar potential, and consequently leads to nonzero
vacuum expectation value for the singlet. Current experimental bounds from the
LHC are imposed on the masses of the composite resonances, while the CP
violating interactions of the pseudo Nambu-Goldstone bosons are tightly
constrained from the measurements of the electric dipole moment of the
electron. We compute the finite temperature corrections to the potential,
incorporating the momentum-dependent form factors in the loop integrals to
capture the effect of the strong dynamics. The impact of the resonances from
the strong sector on the finite temperature potential are exponentially
suppressed. The presence of explicit CP violation leads to strong first-order
phase transition from a false vacuum to the electroweak vacuum where the
pseudoscalar singlet has a non-zero vacuum expectation value. We illustrate
that, as a result of such phase transitions, the production of potentially
observable gravitational waves at future detectors will offer a complementary
avenue to probe the composite Higgs models, distinct from collider experiments.
