The heat flow and temperature structure within and surrounding a magnetic flux tube stored in mechanical equilibrium in a stellar convection zone are considered. The stationary thermal equilibrium state is determined through the analytical solution of a two-dimensional heat diffusion problem for an infinitely long cylinder with different thermal conductivities inside and outside the cylinder, both spatially variable. In the exterior of the cylinder, convective heat transport is approximated in terms of a linear diffusive process, while in its interior convection is assumed to be suppressed and only the much smaller radiative conductivity remains. The results show that, under the conditions prevailing near the bottom of the solar convection zone and in the limit of small cylinder radius, the temperature disturbance (thermal shadow) in the exterior of the insulating cylinder is almost negligible due to the large effiency of convective energy transport. The spatial dependence of the conductivities and the curvature of the external temperature profile lead to a temperature excess in the interior with respect to the undisturbed temperature profile far away from the cylinder. We show that, within the framework of the thin magnetic flux tube approximation, this temperature excess is due to a heating term equal to the negative divergence of the undisturbed radiative heat flow, as suggested earlier by Fan & Fisher (cite{Fan:Fisher:1996}). These results are independent of the treatment of the convective transport in the exterior as long as the stratification is almost adiabatic. The consequences for the storage of magnetic flux in the solar convection zone, brought about by the enhanced buoyancy and caused by the heating effect, are discussed.