Consider the following stochastic differential equation in
$\mathbb{R}^d$:
\begin{eqnarray*}
dX^\varepsilon_t &=& b(X_t^\varepsilon,\xi_{t/\varepsilon})\,dt
+\sqrt{\varepsilon} a(X_t^\varepsilon,\xi_{t/\varepsilon})\,dW_t,\\
X^\varepsilon_0 &=& x_0,
\end{eqnarray*}
where the random environment $(\xi_t)$ is an
exponentially ergodic Markov process, independent of the Wiener process
$(W_t)$, with invariant probability measure $\pi$, and $\varepsilon$ is
some small parameter. In this paper we prove the moderate deviations for
the averaging principle of $X^\varepsilon$, that is, deviations of
$(X^\varepsilon_t)$ around its limit averaging system $(\bar x_t)$ given
by %$\bar{d}$.
%
$d\bar x_t=\bar b(\bar x_t)\,dt$ and $\bar x_0=x_0$ where $\bar
b(x)=\mathbb{E}_\pi(b(x,\cdot))$. More precisely we obtain the large deviation
estimation about
\[
\Bigl(\eta^\varepsilon_t={X^\varepsilon_t-\bar x_t \over
\sqrt{\varepsilon} h(\varepsilon)}\Big)_{t\in[0,1]}
\]
in the space of continuous trajectories, as $\varepsilon$ decreases to 0, where
$h(\varepsilon)$ is some deviation scale satisfying $1\ll
h(\varepsilon)\ll\varepsilon^{-1/2}$. Our strategy will be first to show the
exponential tightness and then the local moderate deviation principle,
which relies on some new method involving a conditional Schilder's
theorem and a moderate deviation principle for inhomogeneous integral
functionals of Markov processes, previously established by the author.