A method for the sensitivity analysis for wall bounded thermal flows including conjugate heat transfer based on the adjoint method is proposed and implemented. In a periodic domain, the flow rate and temperature are kept constant using a uniform source. The solid region is modeled using a volume penalization approach for the velocity and by adapting the thermal diffusivity in the temperature equation. To account for an increased transport due to flow instabilities, time averaging is applied and effective diffusivities are introduced to the adjoint equations. Applied to a channel flow with rectangular obstacles, the sensitivity distribution on the wall surface is computed, showing increased sensitivities especially in regions with high temperature gradients. The obtained sensitivities allow to optimize an initial topology with respect to an objective function. Additionally, the contribution to the sensitivities resulting from different expressions in the primal equations provide insights about the transport mechanisms in the flow. The sensitivities suggest that the surface area should be increased in the regions with high heat fluxes and tend to modify the geometry hindering the occurrence of the flow separation.