  In the studies on pre-recombination early dark energy (EDE), the evolution of
Universe after recombination is usually regarded as ${\Lambda}CDM$-like, which
corresponds that the equation of state of dark energy responsible for current
accelerated expansion is $w=-1$. However, in realistic models, $w$ might be
evolving. We consider the parametrizations of $w$ with respect to the redshift
$z$ in Axion-like EDE and AdS-EDE models, respectively. We performed the Monte
Carlo Markov chain analysis with recent cosmological data, and found that the
bestfit $w(z)$ is compatible with $w_0=-1,w_a=0$ (the cosmological constant)
and the evolution of $w$ is only marginally favored, which so has little effect
on lifting the bestfit value of ${H_0}$.
