Autocatalytic RNA networks confined within coacervate droplets represent minimal systems capable of self-reproduction and compositional variation. While such networks have been studied under static conditions, their behavior under fluctuating environments remains poorly understood. Here, the self-assembly dynamics of the Azoarcus ribozyme is investigated within spermine-polyacrylate coacervates subjected to thermal perturbations. It is shown that temperature acts as a switch for ribozyme catalysis: above a threshold, coacervates support autocatalytic RNA assembly; below it, reaction dynamics are arrested due to diffusion-limited transport. Notably, the switching kinetics are sequence-specific-different ribozyme variants exhibit distinct temperature thresholds for catalytic activity. Reaction rates deviate from Arrhenius behavior inside droplets, highlighting the nontrivial coupling between catalysis and compartmentalization. Finally, it is demonstrated that temperature cycling differentially amplifies ribozyme variants, with selective assembly emerging at intermediate cycling frequencies. These results reveal that fluctuating environments can bias reproduction dynamics in compartmentalized autocatalytic systems, providing a potential route for primitive functional differentiation and selection.