Stochastic temperature distribution should be carefully inspected in the thermal-failure design of heterogeneous solids with unexpected random energy excitations. Stochastic multiscale modeling for these problems involve multiscale and high-dimensional uncertain thermal parameters, which remains limitation of prohibitive computation. In this paper, we propose a multi-modes based constrained energy minimization generalized multiscale finite element method (MCEM-GMsFEM), which can transform the original stochastic multiscale model into a series of recursive multiscale models sharing the same deterministic material parameters by multiscale analysis. Thus, MCEM-GMsFEM reveals an inherent low-dimensional representation in random space, and is designed to effectively reduce the complexity of repeated computation of discretized multiscale systems. In addition, the convergence analysis is established, and the optimal error estimates are derived. Finally, several typical random fluctuations on multiscale thermal conductivity are considered to validate the theoretical results in the numerical examples. The numerical results indicate that the multi-modes multiscale approach is a robust integrated method with the excellent performance.