My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:

Euclidean: $A^2+B^2+C^2=D^2$

Hyperbolic: $\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}−\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos\frac{D}{2}$

Spherical: $\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}+\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos\frac{D}{2}$

where A, B, C are the areas of the "leg-faces" and D is the area of the "hypotenuse-face".

For right-corner simplices in higher Euclidean dimensions, we have that the sum of the squares of the content of leg-simplices equals the square of the content of the hypotenuse-simplex.

I mentioned not knowing the non-Euclidean counterparts of the generalization. After a couple of days of toying, I find these counterparts elusive. (It would probably help if I were better-versed in differential geometry.) So, I turn to MO to ask:

What are the non-Euclidean analogues in higher dimensions?
(I'm particularly interested in the case relating volumes in a hyperbolic 4-simplex.)

While the Pythagorean Theorems for Euclidean Simplices progress in a straightforward manner (just add another leg-content-square), the Pythagorean Theorem for Hyperbolic Tetrahedra already diverges somewhat dramatically from its 2-dimensional counterpart, $\cosh a \cosh b = \cosh c$. So, it's not even clear to me what form the relations would take in general.

Edit to add a couple of references
"The Laws of Cosines for Non-Euclidean Tetrahedra" (PDF) (by me) derives a Law of Cosines for which each non-Euclidean Pythagorean Theorem is a special case. I do not know if these results exist elsewhere in the literature. (BTW: There's a lot of unnecessary equation manipulation shown; I was using this as an opportunity to learn LaTeX. :)

A passage in above jumps into a discussion of "pseudofaces" by making passing reference to the Euclidean case. You can read about Euclidean pseudofaces and how I find them useful here:
"Heron-like Results for [Euclidean] Tetrahedral Volume" (by me).

Edit2: A Special Case

Consider the special case of a right-corner hyperbolic $4$-simplex whose "legs" are congruent right-corner tetrahedra with isosceles right-triangle faces of area $A$; the "hypotenuse" is a regular tetrahedron with equilateral faces of area $D$. (Each face of the simplex's hypotenuse is a hyptonuse-face of one of the simplex's legs; by the Pythagorean Theorem for Hyperbolic Tetrahedra, then, we have $\cos(D/2) = \cos(A/2)^3 - \sin(A/2)^3$.)

Let the volume of each leg-tetrahedron be $L$ and the volume of the hypotenuse-tetrahedron be $H$. We can express these in terms of $A$:

$$L=-\frac{3\sqrt{2}}{4} \int_{0}^{A} \frac{\sqrt{1-\sin\theta}}{3+\sin\theta} \log\frac{1-\sqrt{\sin\theta}}{1+\sqrt{\sin\theta}} \\, \mathrm{d}\theta$$

$$H=-3\int_{0}^{A} \frac{\cos\theta}{\left(3+\sin\theta\right)\sqrt{2+\sin\theta}} \log\frac{1-\sqrt{\sin\theta}}{1+\sqrt{\sin\theta}} \\, \mathrm{d}\theta$$

Note that the derivative of $H$ with respect to $L$ is concisely expressed in terms of $A$:

$$\frac{\mathrm{d}H}{\mathrm{d}L}=2\sqrt{2} \cdot \sqrt{ \frac{1+\sin A}{2+\sin A}}$$

That's about as close as I've gotten to relating $H$ and $L$ "neatly".

Now, it's possible to write series for $H$ and $L$ in terms of $A$, then invert the second series to get a series for $A$ in terms of $L$, then substitute back in to the first series to arrive at a series for $H$ in terms of $L$.

$$H = \frac{4}{2!\\;3!} M^3 + \frac{18}{3!\\;5!}M^5 - \frac{ 918 }{4!\\;7!}M^7 + \frac{24786}{5!\\;9!}M^9 - \frac{ 6018759 }{8 \cdot 6!\\;11!} M^{11} - \frac{ 8233607961 }{80\cdot 7!\\;13!} M^{13} - \cdots $$ where $M := (6L)^{1/3}$. This isn't the relation I'm seeking, but observe that, for infinitesimal $L$, we have $H \approx 2 L$; that is, $H^2 \approx 4L^2 = L^2 + L^2 + L^2 + L^2$, which is the corresponding Pythagorean relation for Euclidean simplices.

I'll close here by mentioning a special-special case: When $A=\pi/2$, we have a simplex whose legs are triply-asymptotic right-corner tetrahedra with leg-faces of area $\pi/2$ and whose hypotenuse is a quadrupally-asymptotice regular tetrahedron with equlateral faces of area $\pi$. The volumes of the components attain significant values:

$$L^\star := \frac{1}{2} \; \Im\left(Li_2\left(\exp\frac{i\pi}{2}\right)\right) = \frac{1}{2} \; \sum_{k=1}^{\infty} \frac{1}{k^2}\sin{\frac{\pi k}{2}} = 0.45798\dots$$

$$H^\star := \Im\left(Li_2\left(\exp\frac{i\pi}{3}\right)\right) = \; \sum_{k=1}^{\infty} \frac{1}{k^2}\sin{\frac{\pi k}{3}} = 1.01494\dots$$

... where $Li_2$ is the dilogarithm.

Here, $L^\star$ is half of Catalan's Constant, and $H^\star$ is also known in the literature. (It is, for instance, the maximum of the Clausen function $\mathrm{Cl}_2$.)

If there's going to be a Pythagorean Theorem for Hypberbolic Simplices, then it must apply to this case, ideally relating these values in the non-Euclidean Pythagorean tradition:

$$\text{function}(H^\star) = \text{combination of related functions}(L_1=L^\star;L_2=L^\star;L_3=L^\star;L_4=L^\star)$$

where the right-hand side is symmetric in the four parameters $L_i$ (representing the volumes of the four legs of the simplex), which are all set equal to $L^\star$. (When the formula is populated with infinitesimal quantities, it should collapse to the Euclidean sum-of-squares relation.) However, while the dilogarithm has many interesting properties, the connection between $H^\star$ and $L^\star$ is not obvious (to me). In a separate MO post, I note a hypergeometric series "similar" to a series for $H^\star$ (what I call "$T(1/2)$" there) that has a direct relation to Catalan's constant (and therefore $L^\star$), but this hasn't yet provided appropriate insights into relating $H^\star$ to $L^\star$ directly.

Have I perhaps lost the forest amid a bunch of trees?