  We present a new type of generalization of the Renyi entropy that follows
naturally from its representation as a thermodynamic quantity. We apply it to
the case of d-dimensional conformal field theories (CFTs) reduced on a region
bounded by a sphere. It is known how to compute their Renyi entropy as an
integral of the thermal entropy of hyperbolic black holes in (d+1)-dimensional
anti-de Sitter spacetime. We show how this integral fits into the framework of
extended gravitational thermodynamics, and then point out the natural
generalization of the Renyi entropy that suggests itself in that light. In
field theory terms, the new generalization employs aspects of the physics of
Renormalization Group (RG) flow to define a refined version of the reduced
vacuum density matrix. For d=2, it can be derived directly in terms of twist
operators in field theory. The framework presented here may have applications
beyond this context, perhaps in studies of both quantum and classical
information theoretic properties of a variety of systems.
