Thermo-mechanical topology optimization requires sensitivity formulations that consistently account for the coupling between heat transfer (via conduction), thermal expansion, and linear elasticity. The adjoint-based treatment of such coupling is well established in the topology optimization literature [1] and has been successfully applied to complex thermo-mechanical design problems, including stress-constrained formulations with temperature-dependent allowable stress [2]. However, fully coupled adjoint sensitivity formulations are less commonly employed in general density-based thermo-mechanical topology optimization problems involving multiple independent design variables. This work presents a coupled adjoint sensitivity formulation for density-based thermo-mechanical topology optimization applicable to both single-material and multi-material designs. The governing problem consists of steady-state heat conduction and linear elastic equilibrium with thermal expansion. The objective is to minimize weighted thermo-mechanical compliance, combining normalized structural and thermal contributions, subject to a global mass fraction constraint. Two independent element-wise design variables are introduced: one governing material existence and one governing material selection. Both design variables influence the thermal and mechanical governing equations, resulting in distinct adjoint sensitivity equations associated with material existence and material selection that must be treated consistently. An adjoint-based formulation is derived that retains all thermo-mechanical coupling terms and remains valid for commonly used differentiable multi-material interpolation strategies, without requiring modification of the sensitivity equations. Numerical examples demonstrate stable convergence and physically meaningful thermo-mechanical designs for both single-material and multi-material cases. The results highlight the importance of sensitivity consistency in density-based thermo-mechanical topology optimization with multiple design variables and provide a robust baseline for density-based multidisciplinary thermo-mechanical design.