In this paper, we present the adaptation of the terminal component learning-based model predictive control (TC-LMPC) architecture for autonomous racing to the Formula Student Driverless (FSD) context. We test the TC-LMPC architecture, a reference-free controller that is able to learn from previous iterations by building an appropriate terminal safe set and terminal cost from collected trajectories and input sequences, in a vehicle simulator dedicated to the FSD competition. One major problem in autonomous racing is the difficulty in obtaining accurate highly nonlinear vehicle models that cover the entire performance envelope. This is more severe as the controller pushes for incrementally more aggressive behavior. To address this problem, we use offline and online measurements and machine learning (ML) techniques for the online adaptation of the vehicle model. We test two sparse Gaussian process regression (GPR) approximations for model learning. The novelty in the model learning segment is the use of a selection method for the initial training dataset that maximizes the information gain criterion. The TC-LMPC with model learning achieves a 5.9 s reduction (3%) in the total 10-lap FSD race time.