  Higher-abelian gauge theories associated with Cheeger-Simons differential
characters are studied on compact manifolds without boundary. The paper
consists of two parts: First the functional integral formulation based on zeta
function regularization is revisited and extended in order to provide a general
framework for further applications. A field theoretical model - called extended
higher-abelian Maxwell theory - is introduced, which is a higher-abelian
version of Maxwell theory of electromagnetism extended by a particular
topological action. This action is parametrized by two non-dynamical harmonic
forms and generalizes the $\theta$-term in usual gauge theories. In the second
part the general framework is applied to study the topological Casimir effect
in higher-abelian gauge theories at finite temperature at equilibrium. The
extended higher-abelian Maxwell theory is discussed in detail and an exact
expression for the free energy is derived. A non-trivial topology of the
background space-time modifies the spectrum of both the zero-point fluctuations
and the occupied states forming the thermal ensemble. The vacuum (Casimir)
energy has two contributions: one related to the propagating modes and the
second one related to the topologically inequivalent configurations of
higher-abelian gauge fields. In the high temperature limit the leading term is
of Stefan-Boltzmann type and the topological contributions are suppressed. With
a particular choice of parameters extended higher-abelian Maxwell theories of
different degrees are shown to be dual. On the $n$-dimensional torus we provide
explicit expressions for the thermodynamic functions in the low- and high
temperature regimes, respectively. Finally, the impact of the background
topology on the two-point correlation function of a higher-abelian variant of
the Polyakov loop operator is analyzed.
