More specifically, for condensed anima, the tangent $\infty$-topos is simply the $\infty$-category of pairs $(X,A)$ where $X$ is a condensed anima and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a hypercomplete sheaf of spectra on the site of condensed sets over $X$. This is definitely a very interesting structure. This whole concept of $6$-functor formalisms is very much about such categories. Working with torsion coefficients, and allowing only "relatively discrete" coefficients, I've developed something along those lines in Etale cohomology of diamonds. Something even closer is in Chapter VII The geometrization of the local Langlands correspondence (link should be active in a few days), where we restrict to the solid objects in $\mathcal D(\mathrm{Cond}_{/X},\mathbb Z_\ell)$. A critical role is then played by the left adjoint $f_\natural$ to pullback $f^\ast$. These do not exist in any classical setup, but have excellent formal properties. In fact, one gets a variant of a $6$-functor formalism where homology and cohomology are now on equal footing again (and arguably homology is even more primitive, again): The pullback functor $f^\ast$ admits a left adjoint $f_\natural$ ("homology") and a right adjoint $Rf_\ast$ ("cohomology"), both of which commute with any pullback. Moreover, Poincare duality holds for proper smooth maps. So yes, there's something interesting about this.

More specifically, for condensed anima, the tangent $\infty$-topos is simply the $\infty$-category of pairs $(X,A)$ where $X$ is a condensed anima and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a hypercomplete sheaf of spectra on the site of condensed sets over $X$. This is definitely a very interesting structure. This whole concept of $6$-functor formalisms is very much about such categories. Working with torsion coefficients, and allowing only "relatively discrete" coefficients, I've developed something along those lines in Etale cohomology of diamonds. Something even closer is in Chapter VII of Geometrization of the local Langlands correspondence (link should be active in a few days is active), where we restrict to the solid objects in $\mathcal D(\mathrm{Cond}_{/X},\mathbb Z_\ell)$. A critical role is then played by the left adjoint $f_\natural$ to pullback $f^\ast$. These do not exist in any classical setup, but have excellent formal properties. In fact, one gets a variant of a $6$-functor formalism where homology and cohomology are now on equal footing again (and arguably homology is even more primitive, again): The pullback functor $f^\ast$ admits a left adjoint $f_\natural$ ("homology") and a right adjoint $Rf_\ast$ ("cohomology"), both of which commute with any pullback. Moreover, Poincare duality holds for proper smooth maps. So yes, there's something interesting about this.

$\require{AMScd}$

Edit: Thanks to the David's for their enlightening comments! Now I understand that the part with the differentials is really only in the examples, not in the "differential cohomology hexagon". For my convenience, let me reformulate this hexagon in my own language.

Say $X$ is a condensed anima, and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a sheaf of spectra on condensed sets over $X$. (Or take $X$ a scheme, and $A\in \mathcal D(X_{\mathrm{proét}},\mathbb Z_\ell)$, or $X$ a small v-stack and $A\in \mathcal D(X_v,\mathbb Z_\ell)$, or...) Let $\pi$ denote the projection from the site of $X$ to the (pro-étale) site of the point. Then pullback $\pi^\ast$ has a right adjoint $R\pi_\ast$ ("cohomology") and a left adjoint that I will denote $\pi_\natural$ ("homology").

In particular, we get condensed spectra $R\pi_\ast A$ (=$R\Gamma(X,A)$), the cohomology of $A$, and $\pi_\natural A$, the homology of $A$. By adjunction, we get maps $$ \pi^\ast R\pi_\ast A\to A\to \pi^\ast \pi_\natural A. $$ Let $\overline{A}=\mathrm{cofib}(\pi^\ast R\pi_\ast A\to A)$ and $\tilde{A}=\mathrm{fib}(A\to \pi^\ast \pi_\natural A)$. Then there is a pullback square $$\begin{CD} A @>>> \overline{A}\\@VVV @VVV\\ \pi^\ast \pi_\natural A @>>> \pi^\ast \pi_\natural \overline{A} \end{CD} $$ and a pushout square $$\begin{CD} \pi^\ast R\pi_\ast \tilde{A} @>>> \pi^\ast R\pi_\ast A\\@VVV @VVV\\ \tilde{A} @>>> A \end{CD} $$ Of course, pushout squares and pullback squares are equivalent, but I want to stress that one wants to use them to recover $A$ from "simpler" information. However, to me $\tilde{A}\to A\to \overline{A}$ all feel extremely similar, and I'd regard these squares as simple statements about how to analyze the small difference between them.