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The Hubble tension persists as a challenge in cosmology. Even early darkenergy (EDE) models, initially considered the most promising for alleviatingthe Hubble tension, fall short of addressing the issue without exacerbatingother tensions, such as the $S_8$ tension. Considering that a negative darkmatter (DM) equation of state (EoS) parameter is conducive to reduce the valueof $\sigma_8$ parameter, in this paper, we extend the axion-like EDE model inthis paper by replacing the cold dark matter (CDM) with DM characterized by aconstant EoS $w_{\rm dm}$ (referred as WDM hereafter). We then imposeconstraints on this axion-like EDE extension model, along with three othermodels: the axion-like EDE model, $\Lambda$WDM, and $\Lambda$CDM. Theseconstraints are derived from a comprehensive analysis incorporating data fromthe Planck 2018 cosmic microwave background (CMB), baryon acoustic oscillations(BAO), the Pantheon compilation, as well as a prior on $H_0$ (i.e.,$H_0=73.04\pm1.04$, based on the latest local measurement by Riess et al.) anda Gaussianized prior on $S_8$ (i.e., $S_8=0.766\pm0.017$, determined throughthe joint analysis of KID1000+BOSS+2dLenS). We find that although the new modelmaintains the ability to alleviate the Hubble tension to $\sim$ 1.4$\sigma$, itstill exacerbate the $S_8$ tension to a level similar to that of the axion-likeEDE model.

Restoring cosmological concordance with axion-like early dark energy and dark matter characterized by a constant equation of state?