This contribution is concerned with the well-posedness and homogenization of an ordinary differential equation (ODE) of Arrhenius-type coupled with a doubly nonlinear parabolic partial differential equation (PDE) with rapidly oscillating coefficients and taking into account disparate diffusion-reaction time scales, including regularly as well as singularly perturbed problems. The ODE-PDE system is spatially dependent and is subjected to Robin-type boundary conditions. Such problems are used to model a variety of processes and phenomena such as combustion and exothermal chemical reactions. We will have a special look at the questions of the existence, uniqueness, boundedness, and the asymptotic limit of the microscale problem by applying the two-scale convergence and unfolding method. A numerical example illustrates both the expected behavior of the approximated solutions as well as the capability of the proposed upscaled models.