  We use the cross-correlation power spectrum of the integrated Sachs-Wolfe
(ISW) effect in the cosmic microwave background (CMB) temperature anisotropy
and galaxy fluctuations to probe the physics of late-time cosmic acceleration.
For this purpose, we focus on three models of dark energy that belong to a
sub-class of Horndeski theories with the speed of gravity equivalent to that of
light: Galileon Ghost Condensate (GGC), Generalized Cubic Covariant Galileon
(GCCG), and K-mouflage. In the GGC and GCCG models, the existence of
cubic-order scalar self-interactions allows a possibility for realizing
negative ISW-galaxy cross-correlations, while the K-mouflage model predicts a
positive correlation similar to the $\Lambda$-cold-dark-matter (LCDM) model. In
our analysis, we fix the parameters of each model to their best-fit values
derived from a baseline likelihood analysis with observational data from CMB,
baryon acoustic oscillations, and supernovae type Ia. Then we fit those
best-fit models to the ISW-galaxy cross-correlation power spectrum extracted
from a collection of photometric redshift surveys. We find that both GGC and
GCCG best-fit models degrade the fit to the ISW-galaxy cross-correlation data
compared to LCDM best-fit model. This is attributed to the fact that, for their
best-fit values constrained from the baseline likelihood, the cubic-order
scalar self-interaction gives rise to suppressed ISW tails relative to
$\Lambda$CDM. The K-mouflage best-fit model is largely degenerate with the LCDM
best-fit model and has a positively correlated ISW-galaxy power close to that
of LCDM.
