The result for $q = 0$ is clear from the definition of $\mathcal{F}^a$. Let $\mathcal{C} = (\Sch/S)_\tau$, or $\mathcal{C} = S_\etale$, or $\mathcal{C} = S_{Zar}$. \medskip\noindent We are going to apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} with $\mathcal{F} = \mathcal{F}^a$, $\mathcal{B} \subset \Ob(\mathcal{C})$ the set of affine schemes in $\mathcal{C}$, and $\text{Cov} \subset \text{Cov}_\mathcal{C}$ the set of standard affine $\tau$-coverings. Assumption (3) of the lemma is satisfied by Lemma \ref{lemma-standard-covering-Cech-quasi-coherent}. Hence we conclude that $H^p(U, \mathcal{F}^a) = 0$ for every affine object $U$ of $\mathcal{C}$. \medskip\noindent Next, let $U \in \Ob(\mathcal{C})$ be any separated object. Denote $f : U \to S$ the structure morphism. Let $U = \bigcup U_i$ be an affine open covering. We may also think of this as a $\tau$-covering $\mathcal{U} = \{U_i \to U\}$ of $U$ in $\mathcal{C}$. Note that $U_{i_0} \times_U \ldots \times_U U_{i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ is affine as we assumed $U$ separated. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-spectral-sequence-application} and the result above we see that $$ H^p(U, \mathcal{F}^a) = \check{H}^p(\mathcal{U}, \mathcal{F}^a) = H^p(U, f^*\mathcal{F}) $$ the last equality by Cohomology of Schemes, Lemma \ref{coherent-lemma-cech-cohomology-quasi-coherent}. In particular, if $S$ is separated we can take $U = S$ and $f = \text{id}_S$ and the proposition is proved. We suggest the reader skip the rest of the proof (or rewrite it to give a clearer exposition). \medskip\noindent Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ on $S$. Choose an injective resolution $\mathcal{F}^a \to \mathcal{J}^\bullet$ on $\mathcal{C}$. Denote $\mathcal{J}^n|_S$ the restriction of $\mathcal{J}^n$ to opens of $S$; this is a sheaf on the topological space $S$ as open coverings are $\tau$-coverings. We get a complex $$ 0 \to \mathcal{F} \to \mathcal{J}^0|_S \to \mathcal{J}^1|_S \to \ldots $$ which is exact since its sections over any affine open $U \subset S$ is exact (by the vanishing of $H^p(U, \mathcal{F}^a)$, $p > 0$ seen above). Hence by Derived Categories, Lemma \ref{derived-lemma-morphisms-lift} there exists map of complexes $\mathcal{J}^\bullet|_S \to \mathcal{I}^\bullet$ which in particular induces a map $$ R\Gamma(\mathcal{C}, \mathcal{F}^a) = \Gamma(S, \mathcal{J}^\bullet) \longrightarrow \Gamma(S, \mathcal{I}^\bullet) = R\Gamma(S, \mathcal{F}). $$ Taking cohomology gives the map $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ which we have to prove is an isomorphism. Let $\mathcal{U} : S = \bigcup U_i$ be an affine open covering which we may think of as a $\tau$-covering also. By the above we get a map of double complexes $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{J}|_S) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}). $$ This map induces a map of spectral sequences $$ {}^\tau\! E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a)) \longrightarrow E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) $$ The first spectral sequence converges to $H^{p + q}(\mathcal{C}, \mathcal{F})$ and the second to $H^{p + q}(S, \mathcal{F})$. On the other hand, we have seen that the induced maps ${}^\tau\! E_2^{p, q} \to E_2^{p, q}$ are bijections (as all the intersections are separated being opens in affines). Whence also the maps $H^n(\mathcal{C}, \mathcal{F}^a) \to H^n(S, \mathcal{F})$ are isomorphisms, and we win.