  Geometry of the universe has always intrigued mathematicians and
cosmologists. Recent results from the Wilkinson Microwave Anisotropy Project
(WMAP) indicate that the visible universe is incredibly flat. This apparent
flatness could be due to the fact that only a small part of the universe is
visible, thus indicating that the geometry of the universe is still an open and
an interesting problem in cosmology. Assuming a profound connection between
Friedmann, Robertson, Walker (FRW) geometries and thermodynamics, we construct
a parameter free exploratory model that allows us to predict the geometry of
the universe by thermodynamic arguments. The key parameters in this model are
the concept of global equation of state and the concept of gravitational
temperature. By comparing the equal time expansion of the Green function for
the massless conformal scalar field in background FRW geometry with the thermal
Green function in Minkowski space-time, we define the gravitational
temperature. We also give the protocol for determining the global equation of
state for a given local equation of state. Using a local equation of state that
covers a wide range of physically acceptable cases,
P={\alpha}{\rho},{\alpha}>0, and within the context of FRW thermodynamics, we
predict that the geometry of the visible universe is Lobachevskian (open),
albeit being very close to flat. This is consistent with the WMAP data, which
indicates that the universe may deviate from flatness by as much as 1%. We also
discuss the self-consistency of this suggestive model along with its possible
connections with nonextensive thermodynamics and black hole thermodynamics.
