This paper proposes a Barrier Function-based Adaptive Super-Twisting Algorithm (BF-ASTA) for Tensioned Leg Platform (TLP)-based floating offshore wind turbines (FOWT). The barrier function allows the gain adaptation to ensure, in a finite time, the convergence of the system states to their references in a predefined range with two main advantages: it does not need the knowledge of the upper bound of perturbations and it does not overestimate the control gain. The proposed BF-ASTA is designed based on a control-oriented model of the 5MW TLP-based FOWT and is validated on the high-fidelity code OpenFAST. The simulation results show the capacity of the controller to regulate the rotor speed at its nominal value and to reduce the platform pitch angle while less blade stresses compared to the proportional integral gain-scheduling (GSPI) controller.