  For a massive scalar field with a general curvature coupling parameter, we
investigate the finite temperature contributions to the Hadamard function and
to the charge and current densities in the geometry of a magnetic flux carrying
generalized cosmic string embedded in $(D+1)$-dimensional locally AdS spacetime
with a compactified spatial dimension. For $D=4$, the geometry on the AdS
boundary, in the context of the AdS/CFT duality, corresponds to a cosmic string
as a linear defect, compactified along its axis. In contrast to the case of the
Minkowski bulk, the upper bound on the chemical potential does not depend on
the field mass and is completely determined by the length of compact dimension
and by the enclosed magnetic flux. The only nonzero components correspond to
the charge density, to the azimuthal current and to the current along the
compact dimension. They are periodic functions of magnetic fluxes with the
period equal to the flux quantum. The charge density is an odd function and the
currents are even functions of the chemical potential. At high temperatures the
influence of the gravitational field and topology on the charge density is
subdominant and the leading term in the corresponding expansion coincides with
that for the charge density in the Minkowski spacetime. The current densities
are topology-induced quantities and their behavior at high temperatures is
completely different with the linear dependence on the temperature. At small
temperatures and for chemical potentials smaller than the critical value, the
thermal expectation values are exponentially suppressed for both massive and
massless fields.
