Suppose that $\mathcal{T}$ is a topos, and let $G$ be a topological group. If you want there to be a "universal $G$-topos" over $\mathcal{T}$, then you need a geometric morphism $f : \mathcal{T} \to \mathbf{Cont}(G)$, where $\mathbf{Cont}(G)$ is the topos of sets with a continuous $G$-action. There is also a geometric morphism $p : \mathbf{Sets} \to \mathbf{Cont}(G)$, with $p^*$ the forgetful functor. The universal $G$-topos over $\mathcal{T}$ is then the pullback of $p$ along $f$.

For example, if $\mathcal{T}=\mathbf{Sh}(S^1)$, then there is a geometric morphism $f : \mathbf{Sh}(S^1) \to \mathbf{Cont}(\mathbb{Z})$, where $\mathbb{Z}$ is the discrete group of integers under addition. Here $f^*$ sends the $\mathbb{Z}$-set $\mathbb{Z}$ to the sheaf corresponding to the projection $\mathbb{R} \to S^1, t \mapsto e^{it}$ (this completely determines $f^*$ because $f^*$ preserves colimits). If you then compute the pullback of $p : \mathbf{Sets} \to \mathbf{Cont}(\mathbb{Z})$ along $f$, you get the "universal $\mathbb{Z}$-topos" over $\mathbf{Sh}(S^1)$, which is given by $\mathbf{Sh}(\mathbb{R})$.

In your setting, if $X$ is a variety over a field $k$, then the morphism of schemes $X \to \mathrm{Spec}(k)$ induces a geometric morphism between the small étale toposes $X_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. Further you can prove that 
$\mathrm{Spec}(k)_\mathrm{\acute{e}t} \simeq \mathbf{Cont}(G_k)$, where $G_k$ is the absolute Galois group of $k$ (with its usual topology).

So in this case we do have a geometric morphism $X_\mathrm{\acute{e}t} \to \mathbf{Cont}(G_k)$, so it makes sense to talk about the universal $G_k$-topos over $X_\mathrm{\acute{e}t}$. 

I don't know precisely how to prove that the universal $G_k$-topos over $X_\mathrm{\acute{e}t}$ is equivalent to $(X_{\bar{k}})_\mathrm{\acute{e}t}$. There are two strategies:

 1. The point $p : \mathbf{Sets} \to \mathbf{Cont}(G_k)$ agrees with the natural geometric morphism $\mathrm{Spec}(\bar{k})_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. So if you show that this pseudopullback of small étale toposes (in the category of toposes) is computed by taking the pullback of the schemes, then this finishes the proof. EDIT: in [this](https://mathoverflow.net/questions/350310/when-is-the-%C3%A9tale-topos-of-a-fibre-product-the-fibre-product-of-%C3%A9tale-toposes) MathOverflow question it is claimed that this holds in the relevant case, because $\mathrm{Spec}(\bar{k})$ is qcqs and pro-étale over $\mathrm{Spec}(k)$. 
 2. If $X$ is a topological space with an action of a discrete group $G$, then the universal $G$-topos over the topos of $G$-equivariant sheaves $\mathbf{Sh}_G(X)$ is given by $\mathbf{Sh}(X)$. Maybe one can show the following more general statement: that if $G$ is a topological group acting continuously on a topos $\mathcal{E}$, then the universal $G$-topos over $\mathbf{Sh}_G(\mathcal{E})$ is given by $\mathcal{E}$. I don't know how to make these defintions precise though.

EDIT: 
Here is a "topos-theoretic proof" of the property $\Gamma_X(F) = \Gamma(G_K,\Gamma_{X_\bar{k}}(q^*F))$ that you mentioned. I use here the name $q$ for the projection $(X_\bar{k})_\mathrm{\acute{e}t} \to X_\mathrm{\acute{e}t}$. I'm not sure if this will be helpful to you, but I'll add it for future reference.

Consider the geometric morphisms $p : \mathrm{Spec}(\bar{k})_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$ and $f : X_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. The pullback of $p$ along $f$ is $q$ as above, and I'll write $g$ for the pullback of $f$ along $p$.

I claim $f$ is a tidy geometric morphism. Because $p$ is an open surjection, it is enough to show that $g$ is tidy (Johnstone's Elephant, C.5.1.7). I added a proof that $g$ is tidy [here](https://mathoverflow.net/a/407756/). So $f$ is tidy as well.

Since $f$ is tidy, the Beck—Chevalley condition $p^*f_* \simeq g_*q^*$ holds (Johnstone's Elephant, C.3.4.11). Applying this to a sheaf $F$ gives
$p^*f_*F \simeq g_*(q^*F) = \Gamma_{X_\mathrm{\acute{e}t}}(q^*F)$. This means that $f_*F$ has as underlying set precisely $\Gamma_{X_\mathrm{\acute{e}t}}(q^*F)$, and then there is a certain $G_k$-action on it. Taking the fixed points under the $G_k$-action amounts to taking global sections $\Gamma_k$ of the sheaf. So we get:
$\Gamma(G_K,\Gamma_{X_\bar{k}}(q^*F)) = \Gamma_{k}(f_*F) = \Gamma_X(F).$