  In this work we prove that the maximally symmetric vacuum solutions of
General Relativity emerge from the geometric structure of statistical mechanics
and thermodynamic fluctuation theory. To present our argument, we begin by
showing that the pseudo-Riemannian structure of the Thermodynamic Phase Space
is a solution to the vacuum Einstein-Gauss-Bonnet theory of gravity with a
cosmological constant. Then, we use the geometry of equilibrium thermodynamics
to demonstrate that the maximally symmetric vacuum solutions of Einstein's
Field Equations -- Minkowski, de-Sitter and Anti-de-Sitter spacetimes --
correspond to thermodynamic fluctuations. Moreover, we argue that these might
be the only possible solutions that can be derived in this manner. Thus, the
results presented here are the first concrete examples of spacetimes
effectively emerging from the thermodynamic limit over an unspecified
microscopic theory without any further assumptions.
