Real-time optimization can play a key role for improving the performance of industrial processes, but it becomes challenging in cases where the process characteristics are uncertain, particularly when the uncertain characteristics impact safety-related constraints. In this study, we present an adaptive and explorative real-time optimization framework that can effectively learn the characteristics of an industrial refrigeration plant through externally driven changes in cooling load targets and through self-exploration. We utilize Gaussian processes (GP) to facilitate learning unknown compressor characteristics of the plant and we leverage the uncertainty quantification of the GP to drive exploration using a weighted sum term in the objective function for real-time optimization. Furthermore, the uncertainty information is used to probabilistically enforce the maximum total power consumption constraint for the compressors with high confidence at all times. Our simulated experiments demonstrate that the proposed approach safely enhances the energy efficiency of the refrigeration process, closely approximating the performance of a best-case solution that has complete information about the plant performance characteristics. We also demonstrate the impact of varying the exploration term in the solution and how the uncertainty of plant behaviour is reduced even in the absence of cooling load target changes.