Efficiently identifying sleep stages is crucial for unraveling the intricacies of sleep in both preclinical and clinical research. The labor-intensive nature of manual sleep scoring, demanding substantial expertise, has prompted a surge of interest in automated alternatives. Sleep studies in mice play a significant role in understanding sleep patterns and disorders and underscore the need for robust scoring methodologies. In response, this study introduces LG-Sleep, a novel subject-independent deep neural network architecture designed for mice sleep scoring through electroencephalogram (EEG) signals. LG-Sleep extracts local and global temporal transitions within EEG signals to categorize sleep data into three stages: wake, rapid eye movement (REM) sleep, and non-rapid eye movement (NREM) sleep. The model leverages local and global temporal information by employing time-distributed convolutional neural networks to discern local temporal transitions in EEG data. Subsequently, features derived from the convolutional filters traverse long short-term memory blocks, capturing global transitions over extended periods. Crucially, the model is optimized in an autoencoder-decoder fashion, facilitating generalization across distinct subjects and adapting to limited training samples. Experimental findings demonstrate superior performance of LG-Sleep compared to conventional deep neural networks. Moreover, the model exhibits good performance across different sleep stages even when tasked with scoring based on limited training samples.