Replanning in temporal logic tasks is extremely difficult during the online execution of robots. This study introduces an effective path planner that computes solutions for temporal logic goals and instantly adapts to non-static and partially unknown environments. Given prior knowledge and a task specification, the planner first identifies an initial feasible solution by growing a sampling-based search tree. While carrying out the computed plan, the robot maintains a solution library to continuously enhance the unfinished part of the plan and store backup plans. The planner updates existing plans when meeting unexpected obstacles or recognizing flaws in prior knowledge. Upon a high-level path is obtained, a trajectory generator tracks the path by dividing it into segments of motion primitives. Our planner is integrated into an autonomous mobile robot system, further deployed on a multicopter with limited onboard processing power. In simulation and real-world experiments, our planner is demonstrated to swiftly and effectively adjust to environmental uncertainties.