Properties of the internal language of the category of sheaves. Consider a simple case of sheaves on some topological space $X$, $\operatorname{Sh}(X)$ (recall that a sieve on $U$ is covering iff its $\operatorname{sup}$ is $U$). All of these are Grothendieck toposes but clearly not all of them are equivalent.
Is there an overview of what properties of $\operatorname{Sh}(X)$ are induced by the properties of $X$? The properties I am interested in are the logical principles that hold in the internal logic. Of course the properties must necessarily be the properties of the locale of opens $\mathcal{O}(X)$.
The elephant (Johnstone) has a list of some properties induced by the properties of $\mathcal{O}(X)$, but most of these are about the existence of geometric morphisms, which of course in some cases allow one to define some modalities in the internal language, but I would also like to know if there are some formulas of ordinary higher-order logic that are valid.
For example, suppose $X$ is compact. Is there a useful or interesting logical principle that holds in $\operatorname{Sh}(X)$ but does not hold in sheaves over non-compact spaces? Perhaps some form of choice?
 A: A topological space (or locale) $X$ is local (i.e. in any covering of $X$ by open subsets at least one such subset is already equal to $X$) iff the internal language of $\mathrm{Sh}(X)$ fulfils the following disjunction property: If $(\phi_i)_{i \in I}$ is an arbitrary family of formulas and $\mathrm{Sh}(X) \models \bigvee_{i \in I} \phi_i$, then there exists an index $i \in I$ such that $\mathrm{Sh}(X) \models \phi_i$.
Also, if $X$ is local, the internal language fulfils the following existence property: If $\mathrm{Sh}(X) \models \exists x : \mathcal{F}. \phi(x)$, then there actually exists a global section $x \in \Gamma(X,\mathcal{F})$ such that $\mathrm{Sh}(X) \models \phi(x)$.
A topological space $X$ is compact iff the internal language of $\mathrm{Sh}(X)$ fulfils the following property: If $I$ is a directed set, $(\phi_i)_{i \in I}$ is a monotone family of formulas (meaning $\mathrm{Sh}(X) \models (\phi_i \Rightarrow \phi_j)$ for $i \preceq j$) and $\mathrm{Sh}(X) \models \bigvee_{i \in I} \phi_i$, then there exists an index $i \in I$ such that $\mathrm{Sh}(X) \models \phi_i$. This follows directly from the order-theoretic characterization of compactness (a space $X$ being compact iff for any monotone family $(U_i)_{i \in I}$ of open sets with $X = \bigcup_i U_i$, there exists $i \in I$ such that $X = U_i$).
As an example, you can use the latter characterization to give an internal proof of lemma 01BB in the Stacks Project (saying that if a filtered colimit $\mathcal{F} = \mathrm{colim}_i \mathcal{F}_i$ of $\mathcal{O}_X$-modules is of finite type on a quasi-compact scheme $X$, then one of the maps $\mathcal{F}_i \to \mathcal{F}$ is an epimorphism) by reducing to the following familiar fact of constructive linear algebra: If a filtered colimit $A = \mathrm{colim}_i A_i$ is finitely generated, one of the maps $A_i \to A$ is surjective.
Unfortunately, the logical principles above are meta properties of the internal language, so I'm not sure this answers your question.
Addendum: An intrinsic characterization of compactness is not possible,
i.e. there cannot exist a formula $\phi$ such that the internal language of the
sheaf topos of a topological space satisfies $\phi$ iff the space is compact:
If a topological space $X$ can be covered by compact subspaces $U_i$, the
formula $\phi$ would be satisfied on each $U_i$ and hence, by the local
character of the internal language, on $X$ as well. If $X$ itself is not
compact, this is a contradiction.
