  A dark energy-like component in the early universe, known as early dark
energy (EDE), is a proposed solution to the Hubble tension. Currently, there is
no consensus in the literature as to whether EDE can simultaneously solve the
Hubble tension and provide an adequate fit to the data from the cosmic
microwave background (CMB) and large-scale structure of the universe. In this
work, we deconstruct the current constraints from the Planck CMB and the
full-shape clustering data of the Baryon Oscillation Spectroscopic Survey
(BOSS) to understand the origin of different conclusions in the literature. We
use two different analyses, a grid sampling and a profile likelihood, to
investigate whether the current constraints suffer from volume effects upon
marginalization and are biased towards some values of the EDE fraction,
$f_\mathrm{EDE}$. We find that $f_\mathrm{EDE}$ allowed by the data strongly
depends on the particular choice of the other parameters of the model and that
several choices of these parameters prefer larger values of $f_\mathrm{EDE}$
than in the Markov Chain Monte Carlo analysis. This suggests that volume
effects are the reason behind the disagreement in the literature. Motivated by
this, we use a profile likelihood to analyze the EDE model and compute a
confidence interval for $f_\mathrm{EDE}$, finding $f_\mathrm{EDE} = 0.072\pm
0.036$ ($68\%$ C.L.). This confidence interval is not subject to volume
effects; thus, our approach yields more robust constraints on EDE and provides
a powerful tool to understand whether EDE is a possible solution to the Hubble
tension.
