UNIVERSAL PRESSURE: Another universal constant?

Not exactly "another" constant, more like the marriage of the familiar two, H and G. Conventionally the Hubble parameter, H, in the empirical v=Hs (where v can be the velocity of a galaxy within a cluster of galaxies that is moving away from an observer on Earth at distance s). The constant, G, of course is the gravitational constant. It has been published that the ratio H/G, where H is derived as an acceleration, so that H/G = -m/r^2, where m is the mass/energy of a sufficiently large region of space with and/or without visible matter of radius r. This region of space is expanding at acceleration H. The ratio has units that suggest pressure (of a sort) -- the background pressure driving accelerated universal expanding space?