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 $\mathrm{Spec}(k)_\mathrm{\acute{e}t} \to \mathrm{Spec}(\bar{k})_\mathrm{\acute{e}t}$. So if you show that pseudopullbacks of small étale toposes (in the category of toposes) are computed by taking the pullback of the schemes, then this finishes the proof.
 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.