Out-of-distribution (OOD) detection has been extensively studied in order to successfully deploy neural networks, in particular, for safety-critical applications. Moreover, performing OOD detection on large-scale datasets is closer to reality, but is also more challenging. Several approaches need to either access the training data for score design or expose models to outliers during training. Some post-hoc methods are able to avoid the aforementioned constraints, but are less competitive. In this work, we propose Generalized ENtropy score (GEN), a simple but effective entropy-based score function, which can be applied to any pre-trained softmax-based classifier. Its performance is demonstrated on the large-scale ImageNet-1k OOD detection benchmark. It consistently improves the average AUROC across six commonly-used CNN-based and visual transformer classifiers over a number of state-of-the-art post-hoc methods. The average AUROC improvement is at least 3.5%. Furthermore, we used GEN on top of feature-based enhancing methods as well as methods using training statistics to further improve the OOD detection performance. The code is available at: https://github.com/XixiLiu95/GEN.