= Solution
Write $\mathcal H=a'/a$, with the prime denoting $\partial_0=c^{-1}\partial_\eta$. The <metric> and <inverse metric> are $g_{\mu\nu}=a^2\eta_{\mu\nu}$ and $g^{\mu\nu}=a^{-2}\eta^{\mu\nu}$. Since $a$ depends only on $x^0$, the <Christoffel symbols> reduce to
$$
\Gamma^\alpha{}_{\beta\gamma}
=\mathcal H\left(\delta^\alpha_\beta\delta^0_\gamma+\delta^\alpha_\gamma\delta^0_\beta-\eta_{\beta\gamma}\eta^{\alpha0}\right).
$$
Consequently
$$
\boxed{\Gamma^0{}_{00}=\mathcal H,\quad\Gamma^0{}_{0i}=0,\quad\Gamma^0{}_{ij}=\mathcal H\delta_{ij},\quad
\Gamma^k{}_{00}=0,\quad\Gamma^k{}_{0i}=\mathcal H\delta^k_i,\quad\Gamma^k{}_{ij}=0.}
$$
For the stated <de Sitter spacetime> scale factor, $\mathcal H=-1/(c\eta)$. The factor $c^{-1}$ is required because the prime is not differentiation with respect to $\eta$ itself.
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