An existing multi-scale FE² finite element framework is extended to steady-state, coupled thermo-mechanical analysis of hyperelastic materials undergoing large deformations. Two formulations are considered: a standard first-order formulation and a thermo-mechanical framework combining second-order mechanical homogenization with a first-order thermal model. The latter is introduced to account for size effects arising from microstructural heterogeneities and thermal expansion. The framework is tested under plane strain conditions for composite microstructures containing either soft or stiff inclusions. The results show that, while the second-order formulation provides a more accurate representation of the RVE fields, the overall macroscale improvement over the classical first-order formulation is problem-dependent. The study demonstrates the practicality of the numerical framework for both approaches in multi-scale thermo-mechanical analysis.