Is the "approaching a value of about 55.8 km/s/Mpc" also the reason why the current consensus seems to be the Heat death (no more potential energy), as opposed to the Big Rip (the acceleration increasing and eventually ripping everything apart from everything else) - and definitely not the Big Crunch (gravity reasserting control and everything going back into a point)? Or am I conflating concepts here? I admit I don't understand why it would be 56.8, I thought the Hubble constant was 70 km/s/Mpc.
The Hubble Parameter (H) varies with time, and with it the Hubble Constant (H₀). The Hubble Constant is just the Hubble Parameter for a given frame of reference (e.g. the Hubble Constant one billion years ago was equal to the Hubble Parameter of that particular point in time (higher than right now, as it is directly proportional to the density of matter in the system), the Hubble Constant in the current frame of reference is equal to the Hubble Parameter in the current frame of reference, in a billion years the Hubble Constant will be equal to the Hubble Parameter of that particular point in time (lower than right now)).
You can look at it is this: Hubble Parameter = √((8πG/3)*ρ) + (Λc²/3 ΩΛ))
*** G = Gravitational Constant, ρ = density of matter, Λ = cosmological constant, c = speed of light (also a constant), ΩΛ = omega lambda sub.
*** the equation also has a subtraction for curvature and scale factors, but that bit can be disregarded because k = 0 in a universe that will become as perfectly flat as possible.
*** In the universe of the far future, ρ (i.e., density of matter in the system) will approach 0 for all intents and purposes.
So you will ultimately have H = √(Λc²/3 ΩΛ)), and if you solve further with estimated ΩΛ values (assuming the cosmological constant and speed of light hold), H will proportionally arrive at roughly √2/3 of the H right now (which is 70 km/s/Mpc, yes). And with that change in the Hubble Parameter (H), you will have a consequent change in the Hubble Constant (H₀).
The Heat Death thing includes everything from universal to remote scales, yes? In a broader sense, the projected development of things being pushed much further away from each other (the Hubble Constant is obviously relevant here) plays a significant role, but in a local sense you also need thermodynamic equilibrium with the dissipation of free energy (which will take very, very, very, very, very long (and life in particular could survive around ultramassive or supermassive black holes until they've disappeared via radiation and loss of angular momentum)).