= Solution
Write $H=\dot a/a$. The nonzero Christoffel symbols for the flat FLRW metric are
$$
\Gamma^t_{ij}=a\dot a\,\delta_{ij},
\qquad
\Gamma^i_{tj}=\Gamma^i_{jt}=H\delta^i_j.
$$
The mixed Einstein tensor is
$$
G_t{}^t=-3H^2,\qquad
G_i{}^j=-\left(2\frac{\ddot a}{a}+H^2\right)\delta_i^j.
$$
For a mixed tensor,
$$
\nabla_\nu G_\mu{}^\nu
=\partial_\nu G_\mu{}^\nu
+\Gamma^\nu_{\nu\lambda}G_\mu{}^\lambda
-\Gamma^\lambda_{\nu\mu}G_\lambda{}^\nu.
$$
For $\mu=t$, direct substitution gives
$$
\begin{aligned}
\nabla_\nu G_t{}^\nu
&=\frac d{dt}(-3H^2)+3H(-3H^2)
-3H\left[-\left(2\frac{\ddot a}{a}+H^2\right)\right]\\
&=-6H\left(\dot H+H^2-\frac{\ddot a}{a}\right)=0.
\end{aligned}
$$
For each spatial $\mu$, homogeneity makes the <partial derivatives> vanish and the remaining diagonal connection terms cancel pairwise. This verifies the <flat FLRW Einstein-tensor divergence> directly.
In vacuum,
$$
G_\mu{}^\nu+\Lambda\delta_\mu^\nu=0.
$$
The $tt$ equation is
$$
H^2=\frac{\Lambda}{3}.
$$
Choosing the positive root gives the expanding <de Sitter scale factor in flat slicing>
$$
\boxed{a(t)=a_0\exp\!\left(\sqrt{\frac{\Lambda}{3}}\,t\right)},
$$
and the spatial equation is then automatically satisfied.
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