The scale-dependent homogenization is applied to a hyperbolic thermoelastic material with two relaxation times, where conductivity and stiffness are wide-sense stationary ergodic random fields. The previously established scaling functions for the Fourier-type conductivity and linear elastic responses are used to describe the trends to scale from the mesoscale statistical volume element level (SVE) to the (representative volume element) RVE level of a deterministic homogeneous continuum. In the case of white-noise type random fields, this finite-size scaling can be quantified via universally appearing stretched exponentials for conductivity and elasticity problems.