Useful invariants of etale topoi not coming from the shape Given a "nice" scheme over a number field or a finite field, etale cohomology (in part because of its functoriality) provides some powerful invariants of the underlying scheme. Sometimes non-abelian invariants like etale fundamental group are also useful. All of these IIUC factor (in the sense of functors) through the shape of the etale topos. Are there any useful invariants of the etale topos of a scheme that are not completely determined by the shape of the topos?
A possibly useful link.
 A: As Harry Gindy as said in the comments, there is a refinement of the notion of shape due to Barwick, Glasman and Haine that contains much more information that just the shape. This is not a pro-space, but a pro-(stratified space), and it essentially remembers the Galois groups of every points of your scheme plus how they are glued together along the specialization poset.
The paper is available on the ArXiv as 

C. Barwick, S. Glasman, P. Haine, Exodromy, arXiv:1807.03281.

This is maybe a bit unfair to call an "invariant" because it remembers the whole étale topos of a qcqs scheme, thanks to what they call "stratified Hochster duality". Recall that classical Hochster duality tells you that taking the specialization order gives an equivalence between spectral spaces and profinite posets. Then in the exodromyy paper they prove

Theorem 10.10 The functor sending a profinite stratified space $\Pi$ to the ∞-topos of constructible sheaves over $\Pi$ is fully faithful. Moreover the essential image consists exactly of the spectral ∞-topoi.

The paper gives a concrete description of spectral ∞-topoi and even better it proves that the étale ∞-topos of every quasi-compact and quasi-separated scheme is spectral. Thus the profinite stratified shape of a qcqs scheme describes completely the étale ∞-topos.
Even better, the stratified shape of a scheme is always 1-truncated, so you can think of it as a category object in Stone spaces. This is a surprisingly concrete object to manipulate, even if it is not necessarily easy to compute (you need, for example, to know the Galois group of every residue field, and the decomposition group for every specialization).
