  In this paper we analyse the vacuum polarization effect associated with the
charged massless scalar field, in the presence of magnetic flux at finite
temperature, in the cosmic string background. We consider a spacetime of an
idealized cosmic string which presents a magnetic field confined in a
cylindrical tube of finite radius. Two different situations are taken into
account in our analysis: (i) a homogeneous field inside the tube and (ii) a
magnetic field proportional to $1/r$. In these two cases, the axis of the
infinitely long tube of radius $R$ coincides with the cosmic string.
Specifically, we calculate the effects produced by the temperature in the
renormalized vacuum expectation value of the square of the charged massless
scalar field, $<\hat{\phi}^{\ast}(x)\hat{\phi}(x)>$. Therefore, in order to
realize these analysis, we calculate the Euclidean Green function associated
with this field in this background.
