What conceivable connection is there between the cosmological constant and radius of the Universe? You probably noticed that the radius of the (observable) Universe in Planck lengths $R$ (about $10^{60}$) and the cosmological constant $\Lambda$ (about $10^{-120}$) in natural units obey approximately:
$$\Lambda R^2 \approx1$$
Suggesting that the amount of dark energy is proportional to the surface area of the cosmological horizon. 
In Hawking's black hole theory the entropy and temperature of a black hole is proportional to its surface area.
This suggests that dark energy should be inversely proportional to the information content of the Universe or proportional to the temperature of the cosmological horizon by the Unruh effect (or seen from the "outside"?)
(Also, it is known that a black hole at a certain size will increase exponentially, as if it has it's own "dark energy" because it is colder than the thermal equilibrium of outer space. Could there be a connection here between how black holes expand and the Universe expands?)
And yet.... the cosmological constant is supposed to be constant. 
So is the relationship merely a coincidence? Are there any current theories that an account for this relationship?
 A: UPDATE: The current theory that accounts for the OP’s relationship is holographic dark energy (HDE). See this paper by Lee et al.  In General Relativity you can consider Λ as as related to the intrinsic torsion or curvature of vacuum. In a quantum universe, dark energy is the energy of the Hawking radiation from a cosmic horizon – the future cosmic event horizon.

To an observer in our universe, a de Sitter universe is in their infinite future. In a de Sitter Universe, the Hubble horizon and Event horizon are coincident. Now, the de Sitter length is equivalent to the event horizon vector $R_h$ to an observer in a dS universe. We assign the de Sitter length $l_Λ=R_h=2=  1/H_0$  and thus de Sitter $Λ=3/4$ in natural units (a bivector), with the energy of the event horizon then being one Planck-energy.
The cosmological constant problem (CCP) can be expressed by observing that the number of degrees of freedom of  dark energy in QFT is much too large to explain the observational data. The CCP can be written as:
$$ρ_P/ρ_Λ =  8πc^3/ℏGΛ=(4/3)πR_h^3=32π/3$$
This approach equates one degree of freedom to one unit volume. The CCP can be overcome by considering the holographic principle which equates the actual number of degrees of freedom of a region to its area not its volume. This is equivalent to stating that the quantum vacuum energy in a region cannot be larger than a black-hole mass of the same size.
The resultant acceleration $a_H$ of the dS event horizon is given by the Unruh relationship: $a_H=1/2=c.H_0.$  as  $T=  (ℏa_H)/(2πck_B )$ This result also aligns with the principle of maximum force in GR i.e. $F_m=(E_H/c^2).a_H=1/2.$ Also, the 'entropic force' at the event horizon is $F_e=-2TS.a_H=-c^4/G$ i.e. Planck force.
From the Freidman equation, the vacuum energy density:
$$p_Λ=  3H_0^2/8πG=Λc^2/8πG$$
Then, extending this answer, the degrees of freedom $N_s$ (dimensionless) on the dS event horizon surface area $A_s$ is:
$$E=2TS=(1/2)N_s k_B T=2$$
$$A_s=4π.R_h^2$$
$$N_S=16π=A_s/L_Planck^2$$
We can thus associate one degree of freedom with one Planck area at the surface of the event horizon, which defines a theory of emergent space. i.e.

‘dark energy is proportional to the surface area of the cosmological horizon’.

Fine. Now what about the OP’s ‘numerological relationship’?
Well, in our universe the Freidman equation is:

As dark energy will be the same density in our infinite future as now:
$$(Λ/3).R_h^2=1$$
You can also think of $Λ=1/l_Λ^2 [1/L^2 ]$  as some kind of cosmic length scale, which, from 2018 Planck data is around $l_Λ=3.08$ GPc. The current particle horizon $R_P$ is $\approx$3 (2.72) times $R_h$ in GPc’s, so if you sub $R_P=2.72\times R_h=5.45$GPc you find  $(Λ/3)R_P^2\approx$1
If you feel this is coincidence, well, HDE solves (Page 5) the cosmic coincidence problem too, as the holographic principle gives an upper bound of N~65 e-folds from the original dS universe.
