For some time now, I'm pondering about a story set up in hyperbolic world. Let me share a few things I came up with and a few I am not sure about:

Absolute unit -- I was thinking of using 10 km as the absolute unit. It's large enough to (hopefully) not make the curvature fatal for human-sized organisms. This would make curvature more of a landscape feature -- it might not be very noticeable in a small town, for example.

Heaven and Hell -- the world is an infinite H2 plane embedded in H3 space, but with normal E1 time (since hyperbolic time axis would cause rapid expansion of space and make infinitely large solid structure impossible). It has a unique "main height level" and is bordered by two equidistant surfaces: "Heaven" at the height of 100 km (10 absolute units) and "Hell" 100 km below. Heaven is a luminescent surface that sends energy to the world, while Hell is a black-hole-like surface that absorbs all energy. Energy that is absorbed by Hell is redistributed and re-radiated from Heaven. While places where Hell is actually exposed are rare, the total amount of energy absorbed by it is infinite, so it can support infinite radiation of Heaven (it takes some SMART redistribution).

Gravity -- here is where I start speculating. I want the world to have standard gravity on the main level. Let's not get into details of what exactly is generating it, it will be 1 standard g on the plane, by fiat. What, then, is gravity in other heights?

In E3, gravity diminishes with square of distance. Strength of gravity of a point source in a given distance corresponds to a surface of a sphere at this distance -- twice the distance means four times bigger surface and so four times smaller gravity.

In the hyperbolic world, the strength of gravity should correspond to the ratio between the area of equidistant surface and the area of the plane, which can be compared by taking a bunch of lines perpendicular to a plane and comparing shapes they cut in the plane and in equidistant surface.

First we need to know what's the length of equidistant curve. If I have two points on a line with distance a, construct perpendicular lines which intersect an equidistant at distance b, then the length of the arc of the equidistant between the two points should be a*cosh(b) -- but I didn't actually calculate this from first principles, only from analogy with spherical geometry, so I don't know if it holds.

Now, for ares on equidistant surfaces, such surface can have an enlarged image of anything on the plane, without distortion. This means that if the length conversion factor is cosh(b), then area conversion factor will be square of that, cosh(b)^2, which can be rewritten as (cosh(2b)+1)/2

This means that the gravity at height h would be smaller by factor of cosh(h)^2. This would hold for both positive and negative height.

The gravity would reduce to 90% at height of 3.3 km, 75% at 5.5 km, 50% at 8.8 km, 10% at 18.1 km and at the lmit distance of 100 km, it would have strength of mere 8.2 x 10^-7 %, making things at these areas effectively weightless.

What I can't completely figure out is how air and water pressure would work. Air pressure can be computed by a differential equation (air density/pressure at given height is dependent on total weight of all air directly above it). The layers of air in hyperbolic world above a given area would be larger the higher they are, but the gravity at such height would also reduce by the same amount, so maybe the air pressure would actually work the same as on Earth? I am not sure...

As for water pressure in oceans, that works out very differently because water is noncompressible and always has the same density. If we assume normal 1 atm on the surface, then pressure in the depth d would be computed by integration like this:

In Euclidean space, regardless of our depth, each layer of water at depth x < d has the same area and same strength of gravity. So at depth d we integrate constant (ro x g) (density of water times gravity) from 0 to d, getting the standard formula of d x ro x g -- plus a constant for atmospheric pressure.

1. Each layer of water in depth x < d has SMALLER area than the area in depth d. The ratio is cosh(x)^2/cosh(d)^2.

2. Weight of each of these layers is diminished by gravity reduction. Gravity at depth x is 1/cosh(x)^2 of the surface gravity, making the weight of each layer of water correspond to 1/cosh(d)^2 -- in other words, the higher layers of water (closer to surface) will be heavier, but they will also have smaller area, making the total weight same for all of them.

So the weight is not just ro x g -- it's ro x g / cosh(d)^2. The deeper you are, the smaller the weight of each individual layer of water above you. So full pressure is d x ro x g / cosh(d)^2 + 1 (for atmospheric pressure on top), i.e. smaller than pressure at the same depth on Earth (or in Euclidean space) by a factor of cosh(d)^2.

This has a surprising consequences: at certain point, as you dive to the hyperbolic ocean, the pressure will reach a local maximum and then it will start to decrease. In this world with absolute length of 10 km, this will happen at the depth of roughly 7700 m, where hyperbolic pressure will be around 448 atm (compared to 771 atm on Earth). Not sure what will happen below this limit -- will the air bubbles and such move down?