We present a dual 5D braneworld picture of a recently suggested model for a microcanonical description of a 4D cosmology driven by a conformal field theory with a large number of quantum fields. The 5D side of the duality relation is represented by a generalized brane induced gravity model in a Schwarzschild-de Sitter bulk. The values of the bulk cosmological and the induced 4D cosmological constants are determined by requiring the absence of conical singularity at the de Sitter horizon of the Euclidean Schwarzschild-de Sitter bulk. Those values belong to the vicinity of the upper bound of a range of admissible values for the cosmological constant. This upper bound is enforced by the 4D CFT and coincides with the natural gravitational cutoff in a theory with many quantum species. The resulting DGP/CFT duality suggests the possibility of a new type of background independent correspondence. A mechanism for inverting the sign of the effective cosmological constant is found, which might reconcile a negative value of the primordial cosmological constant compatible with supersymmetry with the one required by inflationary cosmology.