  We analyze the renormalization group (RG) flow of the temperature independent
term of the entropy in the high temperature limit \beta/a<<1 of a massive field
theory in 3-dimensional spherical spaces M_3 with constant curvature 6/a^2. For
masses lower than 2\pi/\beta, this term can be identified with the free energy
of the same theory on M_3 considered as a 3-dimensional Euclidean space-time.
The non-extensive part of this free energy, S_hol, is generated by the holonomy
of the spatial metric connection. We show that for homogeneous spherical spaces
the holonomy entropy S_hol decreases monotonically when the RG scale flows to
the infrared. At the conformal fixed points the values of the holonomy entropy
do coincide with the genuine topological entropies recently introduced. The
monotonic behavior of the RG flow leads to an inequality between the
topological entropies of the conformal field theories connected by such flow,
i.e. S_top^UV > S_top^IR. From a 3-dimensional viewpoint the same term arises
in the 3-dimensional Euclidean effective action and has the same monotonic
behavior under the RG group flow. We conjecture that such monotonic behavior is
generic, which would give rise to a 3-dimensional generalization of the
c-theorem, along the lines of the 2-dimensional c-theorem and the 4-dimensional
a-theorem. The conjecture is related to recent formulations of the F-theorem.
In particular, the holonomy entropy on lens spaces is directly related to the
topological R\'enyi entanglement entropy on disks of 2-dimensional flat spaces.
