The Einstein equations with a cosmological constant Λ read as:
Rμν−12Rgμν+Λgμν=8πTμν
Therefore,
R−D2R+DΛ=8πT
(D is the number of spacetime dimensions.)
Or,
Λ=8πTD+(D−22D)R
Therefore,
Λ=(D−22D)R0
where R0:=Rvacuum
From the field equations,
Rμν−12Rgμν+(D−22D)R0gμν=8πTμν
Therefore,
8πT=D−22(R0−R)
Now, in the large D limit, the only way T can be saved from diverging is to make R approach R0. This means that in the large D limit, R=R0 everywhere, i.e., the large D limit of General Relativity admits only vacuum solutions.
I am posting this question to confirm if the conclusion I have reached is appropriate because I haven't come across any such claim elsewhere. Also, if this is appropriate then does it denote something interesting or more profound? (Put in other words, can it be associated with some rather known facts or principles?)
PS: I just noticed that in 2-dimensional case (1+1 dimensional case) also, only the vacuum solutions can exist and the cosmological constant also must vanish. (One can verify trivially by putting D=2 in the formula for Λ and T above.)