I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.

The formulation of differential cohomology in certain cohesive $\infty$ topos seems, to me, to be an incredibly powerful generalization of the concepts surrounding classical Chern-Weil theory. I have, for a while, been curious as to what forms differential cohomology may take in topos other than the typical one modeled on cartesian spaces (or differentiable manifolds).

Question: Can someone provide some examples of what form differential cohomology may take in topos modeled on different local data? More precisely, what does the differential cohomology diamond look like in these other topos?

To give the beginning of a possible example, suppose we take as our starting point the p-adics $\mathbb{Q}_p$ instead of the reals and form the cohesive site whose objects are $(\mathbb{Q}_p)^n$, $n\in \mathbb{N}$ and morphisms are $p$-adic smooth maps (for example locally constant when the support is compact). If I'm not mistaken, we should be able to form the differentially cohomology diamond in the resulting $\infty$ topos. Can we identify the components of this diamond with something more familiar from the world of number theory?