  Degree one twisting of Deligne cohomology, as a differential refinement of
integral cohomology, was established in previous work. Here we consider higher
degree twists. The Rham complex, hence de Rham cohomology, admits twists of any
odd degree. However, in order to consider twists of integral cohomology we need
a periodic version. Combining the periodic versions of both ingredients leads
us to introduce a periodic form of Deligne cohomology. We demonstrate that this
theory indeed admits a twist by a gerbe of any odd degree. We present the main
properties of the new theory and illustrate its use with examples and
computations, mainly via a corresponding twisted differential Atiyah-Hirzebruch
spectral sequence.
