Continuously learning new classes without catastrophic forgetting is a challenging problem for on-device acoustic event classification given the restrictions on computation resources (e.g., model size, running memory). To alleviate such an issue, we propose two novel diversity-aware incremental learning method for Spoken Keyword Spotting and Environmental Sound Classification. Our method selects the historical data for the training by measuring the per-sample classification uncertainty. For the Spoken Keyword Spotting application, the proposed RK approach introduces a diversity-aware sampler to select a diverse set from historical and incoming keywords by calculating classification uncertainty. As a result, the RK approach can incrementally learn new tasks without forgetting prior knowledge. Besides, the RK approach also proposes data augmentation and knowledge distillation loss function for efficient memory management on the edge device. For the Environmental Sound Classification application, we measure the uncertainty by observing how the classification probability of data fluctuates against the parallel perturbations added to the classifier embedding. In this way, the computation cost can be significantly reduced compared with adding perturbation to the raw data. Experimental results show that the proposed RK approach achieves 4.2% absolute improvement in terms of average accuracy over the best baseline on Google Speech Command dataset with less required memory. Experimental results on the DCASE 2019 Task 1 and ESC-50 dataset show that our proposed method outperforms baseline continual learning methods on classification accuracy and computational efficiency, indicating our method can efficiently and incrementally learn new classes without the catastrophic forgetting problem for on-device environmental sound classification