  New physics increasing the expansion rate just prior to recombination is
among the least unlikely solutions to the Hubble tension, and would be expected
to leave an important signature in the early Integrated Sachs-Wolfe (eISW)
effect, a source of Cosmic Microwave Background (CMB) anisotropies arising from
the time-variation of gravitational potentials when the Universe was not
completely matter dominated. Why, then, is there no clear evidence for new
physics from the CMB alone, and why does the $\Lambda$CDM model fit CMB data so
well? These questions and the vastness of the Hubble tension theory model space
motivate general consistency tests of $\Lambda$CDM. I perform an eISW-based
consistency test of $\Lambda$CDM introducing the parameter $A_{\rm eISW}$,
which rescales the eISW contribution to the CMB power spectra. A fit to Planck
CMB data yields $A_{\rm eISW}=0.988 \pm 0.027$, in perfect agreement with the
$\Lambda$CDM expectation $A_{\rm eISW}=1$, and posing an important challenge
for early-time new physics, which I illustrate in a case study focused on early
dark energy (EDE). I explicitly show that the increase in $\omega_c$ needed for
EDE to preserve the fit to the CMB, which has been argued to worsen the fit to
weak lensing and galaxy clustering measurements, is specifically required to
lower the amplitude of the eISW effect, which would otherwise exceed
$\Lambda$CDM's prediction by $\approx 20\%$: this is a generic problem beyond
EDE and likely applying to most models enhancing the expansion rate around
recombination. Early-time new physics models invoked to address the Hubble
tension are therefore faced with the significant challenge of making a similar
prediction to $\Lambda$CDM for the eISW effect, while not degrading the fit to
other measurements in doing so.
