We will prove this by induction on $\dim(X)$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$. \medskip\noindent The case $\dim(X) = 0$. In this case the structure morphism $f : X \to \Spec(K)$ is finite. Hence we see that $R^if_*\mathcal{F} = 0$ for $i > 0$, see Proposition \ref{proposition-finite-higher-direct-image-zero}. Thus $H^i_\etale(X, \mathcal{F}) = H^i_\etale(\Spec(K), f_*\mathcal{F})$ by the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray}) and the result is clear. \medskip\noindent The case $\dim(X) = 1$. This is Lemma \ref{lemma-cd-curve-over-field}. \medskip\noindent Assume $d = \dim(X) > 1$ and the proposition holds for finite type affine schemes of dimension $< d$ over fields. By Noether normalization, see for example Varieties, Lemma \ref{varieties-lemma-noether-normalization-affine}, there exists a finite morphism $f : X \to \mathbf{A}^d_K$. Recall that $R^if_*\mathcal{F} = 0$ for $i > 0$ by Proposition \ref{proposition-finite-higher-direct-image-zero}. By the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray}) we conclude that it suffices to prove the result for $\pi_*\mathcal{F}$ on $\mathbf{A}^d_K$. \medskip\noindent Interlude I. Let $j : X \to Y$ be an open immersion of smooth $d$-dimensional varieties over $K$ (not necessarily affine) whose complement is the support of an effective Cartier divisor $D$. The sheaves $R^qj_*\mathcal{F}$ for $q > 0$ are supported on $D$. We claim that $(R^qj_*\mathcal{F})_{\overline{y}} = 0$ for $a = \text{trdeg}_K(\kappa(y)) > d - q$. Namely, by Theorem \ref{theorem-higher-direct-images} we have $$ (R^qj_*\mathcal{F})_{\overline{y}} = H^q(\Spec(\mathcal{O}_{Y, y}^{sh}) \times_Y X, \mathcal{F}) $$ Choose a local equation $f \in \mathfrak m_y = \mathcal{O}_{Y, y}$ for $D$. Then we have $$ \Spec(\mathcal{O}_{Y, y}^{sh}) \times_Y X = \Spec(\mathcal{O}_{Y, y}^{sh}[1/f]) $$ Using Lemma \ref{lemma-strictly-henselian} we get an embedding $$ K(t_1, \ldots, t_a)^{sep}(x) = K(t_1, \ldots, t_a)^{sep}[x]_{(x)}[1/x] \longrightarrow \mathcal{O}_{Y, y}^{sh}[1/f] $$ Since the transcendence degree over $K$ of the fraction field of $\mathcal{O}_{Y, y}^{sh}$ is $d$, we see that $\mathcal{O}_{Y, y}^{sh}[1/f]$ is a filtered colimit of $(d - a - 1)$-dimensional finite type algebras over the field $K(t_1, \ldots, t_a)^{sep}(x)$ which itself has cohomological dimension $1$ by Lemma \ref{lemma-cd-field-extension}. Thus by induction hypothesis and Lemma \ref{lemma-cd-limit} we obtain the desired vanishing. \medskip\noindent Interlude II. Let $Z$ be a smooth variety over $K$ of dimension $d - 1$. Let $E_a \subset Z$ be the set of points $z \in Z$ with $\text{trdeg}_K(\kappa(z)) \leq a$. Observe that $E_a$ is closed under specialization, see Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}. Suppose that $\mathcal{G}$ is a torsion abelian sheaf on $Z$ whose support is contained in $E_a$. Then we claim that $H^b_\etale(Z, \mathcal{G}) = 0$ for $b > a + \text{cd}(K)$. Namely, we can write $\mathcal{G} = \colim \mathcal{G}_i$ with $\mathcal{G}_i$ a torsion abelian sheaf supported on a closed subscheme $Z_i$ contained in $E_a$, see Lemma \ref{lemma-support-in-subset}. Then the induction hypothesis kicks in to imply the desired vanishing for $\mathcal{G}_i$\footnote{Here we first use Proposition \ref{proposition-closed-immersion-pushforward} to write $\mathcal{G}_i$ as the pushforward of a sheaf on $Z_i$, the induction hypothesis gives the vanishing for this sheaf on $Z_i$, and the Leray spectral sequence for $Z_i \to Z$ gives the vanishing for $\mathcal{G}_i$.}. Finally, we conclude by Theorem \ref{theorem-colimit}. \medskip\noindent Consider the commutative diagram $$ \xymatrix{ \mathbf{A}^d_K \ar[rd]_f \ar[rr]_-j & & \mathbf{P}^1_K \times_K \mathbf{A}^{d - 1}_K \ar[ld]^g \\ & \mathbf{A}^{d - 1}_K } $$ Observe that $j$ is an open immersion of smooth $d$-dimensional varieties whose complement is an effective Cartier divisor $D$. Thus we may use the results obtained in interlude I. We are going to study the relative Leray spectral sequence $$ E_2^{p, q} = R^pg_*R^qj_*\mathcal{F} \Rightarrow R^{p + q}f_*\mathcal{F} $$ Since $R^qj_*\mathcal{F}$ for $q > 0$ is supported on $D$ and since $g|_D : D \to \mathbf{A}^{d - 1}_K$ is an isomorphism, we find $R^pg_*R^qj_*\mathcal{F} = 0$ for $p > 0$ and $q > 0$. Moreover, we have $R^qj_*\mathcal{F} = 0$ for $q > d$. On the other hand, $g$ is a proper morphism of relative dimension $1$. Hence by Lemma \ref{lemma-cohomological-dimension-proper} we see that $R^pg_*j_*\mathcal{F} = 0$ for $p > 2$. Thus the $E_2$-page of the spectral sequence looks like this $$ \begin{matrix} g_*R^dj_*\mathcal{F} & 0 & 0 \\ \ldots & \ldots & \ldots \\ g_*R^2j_*\mathcal{F} & 0 & 0 \\ g_*R^1j_*\mathcal{F} & 0 & 0 \\ g_*j_*\mathcal{F} & R^1g_*j_*\mathcal{F} & R^2g_*j_*\mathcal{F} \end{matrix} $$ We conclude that $R^qf_*\mathcal{F} = g_*R^qj_*\mathcal{F}$ for $q > 2$. By interlude I we see that the support of $R^qf_*\mathcal{F}$ for $q > 2$ is contained in the set of points of $\mathbf{A}^{d - 1}_K$ whose residue field has transcendence degree $\leq d - q$. By interlude II $$ H^p(\mathbf{A}^{d - 1}_K, R^qf_*\mathcal{F}) = 0 \text{ for }p > d - q + \text{cd}(K)\text{ and }q > 2 $$ On the other hand, by Theorem \ref{theorem-higher-direct-images} we have $R^2f_*\mathcal{F}_{\overline{\eta}} = H^2(\mathbf{A}^1_{\overline{\eta}}, \mathcal{F}) = 0$ (vanishing by the case of dimension $1$) where $\eta$ is the generic point of $\mathbf{A}^{d - 1}_K$. Hence by interlude II again we see $$ H^p(\mathbf{A}^{d - 1}_K, R^2f_*\mathcal{F}) = 0 \text{ for }p > d - 2 + \text{cd}(K) $$ Finally, we have $$ H^p(\mathbf{A}^{d - 1}_K, R^qf_*\mathcal{F}) = 0 \text{ for }p > d - 1 + \text{cd}(K)\text{ and }q = 0, 1 $$ by induction hypothesis. Combining everything we just said with the Leray spectral sequence $H^p(\mathbf{A}^{d - 1}_K, R^qf_*\mathcal{F}) \Rightarrow H^{p + q}(\mathbf{A}^d_K, \mathcal{F})$ we conclude.