= Solution
This question uses proper cosmological time $t$. The future-directed unit normal is
$$
\boxed{n^\mu=(1,0,0,0),\qquad n_\mu=(-1,0,0,0).}
$$
The <non-null hypersurface projection> removes all normal components. The only relevant <Christoffel symbols> are $\Gamma^0_{ij}=a\dot a\delta_{ij}$. Since the components of $n_\mu$ are constant,
$$
K_{ij}=\nabla_i n_j=-\Gamma^0_{ij}n_0=a\dot a\delta_{ij}=H g_{ij},\qquad K_{0\mu}=0.
$$
Consequently
$$
\boxed{K=3H=3\dot a/a.}
$$
With the printed positive <extrinsic curvature> convention, this is the expansion of the normal comoving congruence. A fixed coordinate cell has proper volume proportional to $a^3$, so $K=d\log(a^3)/dt=\nabla_\mu n^\mu$: the fractional proper-volume expansion rate.
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