Using the differentiable structure, twisted 2-periodic de Rham cohomology is well known, and showing up as the Larget of Chern characters for twisted K-theory. The main motivation of this work is a topological interpretation of two-periodic twisted de Rham cohomology which is generalizable to arbitrary topological spaces and at the same time to arbitrary coefficients.

To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on :

- the construction of the sheaf theory operations in unbounded derived categories

- elements of Verdier duality

- and integration.

The main result is the construction of a functorial periodization associated to a U  (1)-gerbe.

As an application we verify the T -duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology.