Combining control engineering with nonparametric modeling techniques from machine learning allows for the control of systems without analytic description using data-driven models. Most of the existing approaches separate learning, i.e., the system identification based on a fixed dataset, and control, i.e., the execution of the model-based control law. This separation makes the performance highly sensitive to the initial selection of training data and possibly requires very large datasets. This article proposes a learning feedback linearizing control law using online closed-loop identification. The employed Gaussian process model updates its training data only if the model uncertainty becomes too large. This event-triggered online learning ensures high data efficiency and thereby reduces computational complexity, which is a major barrier for using Gaussian processes under real-time constraints. We propose safe forgetting strategies of data points to adhere to budget constraints and to further increase data efficiency. We show asymptotic stability for the tracking error under the proposed event-triggering law and illustrate the effective identification and control in simulation.