  We explore the dependence of the incidence of moderate-luminosity ($L_{X} =
10^{41.9-43.7}$ erg s$^{-1}$) AGNs and the distribution of their accretion
rates on host color at 0.5 < z < 2.5, using deep X-ray data in GOODS fields. We
use extinction-corrected rest-frame U-V colors to divide both AGN hosts and
non-AGN galaxies into red sequence (quiescent), green valley (transition), and
blue cloud (star-forming) populations. We find that both the AGN fraction at
fixed stellar mass and its evolution with redshift are dependent on host
colors. Most notably, red galaxies have the lowest AGN fraction (~5\%) at z~1
yet with most rapid redshift evolution, increasing by a factor of 5 (~24\%) at
z~2. Green galaxies exhibit the highest AGN fraction across all redshifts,
which is most pronounced at z~2 with more than half of them hosting an AGN at
$M_{*} > 10^{10.6} M_{\odot}$. Together with the high AGN fraction in red
galaxies at z~2, this indicates that X-ray AGNs could be important in both
transforming blue galaxies into red ones and subsequently maintaining their
quiescence at high redshift. Furthermore, consistent with low-redshift studies,
we find that the probability of hosting an AGN in the total galaxy population
can be characterized by a universal Eddington ratio ($p(\lambda_{Edd}) \sim
\lambda_{Edd}^{-0.4}$) and a moderate redshift evolution. Yet consistent with
their different AGN fractions, different populations appear to also have
different $p(\lambda_{Edd})$ with red galaxies exhibiting more rapid redshift
evolution than green and blue ones. Evidence for a steeper power law of
$p(\lambda_{Edd})$ in red galaxies is also presented, though larger samples are
needed to confirm. These results suggest that the AGN accretion or the growth
of supermassive black holes is related to their host properties, and may also
influence their hosts in a different mode dependent on the host color.
