Intrinsic topology on the Zariski spectrum In a big topos whose objects are a kind of "space", it sometimes happens that when we define some "set" internally to the topos, the "topology" it automatically acquires coincides with the "correct" or "expected" one for the usual external definition.  For instance, the real numbers object in the topos may be the set of real numbers with its usual "topology" (in whatever sense that topos represents "topology").  For example, this is the case for the Dedekind real numbers in the topos of sheaves on cartesian spaces $\mathbb{R}^n$, and for both Cauchy and Dedekind real numbers (which coincide) in Johnstone's topological topos.
Is there any topos in which this holds for the Zariski spectrum of a commutative ring?  That is, if I start with an ordinary external commutative ring, map it into the topos as a "discrete" object, and then construct the object of prime ideals (or maybe filters -- whatever makes the most sense constructively) of that internal ring object, does the automatic intrinsic "topology" on the resulting object ever coincide with the usual Zariski topology?
One might naturally guess some topos of sheaves on algebraic spaces.  But I don't think it's impossible that Johnstone's topological topos might also work; despite comments elsewhere that convergence isn't useful in the Zariski topology, Wikipedia tells me that the Zariski topology of a commutative Noetherian ring is a sequential topological space, and sequential spaces embed fully-faithfully in the topological topos.
 A: New answer, following up on Zhen Lin's comment for the "good" definition of prime filter. Sorry for the confusion!
A prime filter in a ring $A$ is a multiplicatively closed subset $S$ of $A$ (containing $1$) such that if $a+b\in S$, then $a\in S$ or $b\in S$. This definition makes sense internally in a topos. And it is almost a tautology that the category of sheaves on $\mathrm{Spec}(A)$ is precisely the classifying topos for prime filters in $A$ (where the universal prime filter is the sheaf of units of the structure sheaf).
In particular, if $X$ is any topological space, then continuous morphisms from $X$ to $\mathrm{Spec}(A)$ are equivalent to prime filters on the constant sheaf on $A$ on $X$.
In particular, working on the big site of topological spaces, the space of prime filters on the constant sheaf on $A$ gives precisely the topological space $\mathrm{Spec}(A)$.
(You could also work in pyknotic sets instead, and get the pyknotic space associated to $\mathrm{Spec}(A)$. Or work in Johnstone's topos, and get the sequential space...)
A: Let me give the condensed perspective: Regarding $A$ as a discrete condensed ring, I think the structure of the "internal spectrum" is codified by the functor that takes any extremally disconnected profinite set $S$ to the poset of sheaves of prime ideals in the constant sheaf on $A$ over $S$. (One could forget the poset structure and regard it only as a condensed set. I will comment below what structure this remembers.) Here, a "sheaf of prime ideals" is defined to be a sheaf of ideals $I$ of $A$ together with a sheaf of multiplicative subsets $M$ of $A$ such that the map $I\sqcup M\to A$ is an isomorphism (of sheaves of sets); I hope this is the correct way to talk about "internal prime ideals"?
I claim that this is the "correct" answer to this question. Recall that Makkai's conceptual completeness theorem as explained by Lurie in his course on categorical logic, or by Barwick-Glasman-Haine in their work on exodromy, gives a fully faithful embedding of the category of coherent locales (aka spectral spaces) to condensed posets. In one direction, this takes any coherent locale to the condensed category of points.
Summary: The spectrum of a ring is naturally a spectral space, i.e. coherent locale, so determined by its condensed poset of points. This is precisely the spectrum of $A$ as constructed internally in condensed sets.
Addendum: If one forgets the poset structure and only looks at the condensed set of prime ideals, one actually ends up getting a condensed set that is representable by a profinite set, which is precisely $\mathrm{Spec}(A)$ with its constructible topology.
