T0 censorship of early dark energy and AdS vacua

Present-day temperature $T_0$ of cosmic microwave background has been precisely measured by the FIRAS experiment. We identify why the early dark energy (EDE) (non-negligible around matter-radiation equality) scenario is compatible with the FIRAS result, while lifting the Hubble constant $H_0$. We perform Monte Carlo Markov Chain analysis to confirm our observations. We also present an $\alpha$-attractor Anti-de Sitter (AdS) model of EDE. As expected, the existence of an AdS phase near recombination can effectively result in $H_0\sim 73$km/s/Mpc at 1$\sigma$ region in the bestfit model.