The analysis of compositional longitudinal data, particularly in microbiome time-series, is a challenging task due to its high-dimensional, sparse, and compositional nature. In this paper, we introduce a novel Gaussian process (GP) prior variational autoencoder for longitudinal data analysis with a multinomial likelihood (MNLVAE) that is specifically designed for compositional time-series analysis. Our generative deep learning model captures complex interactions among microbial taxa while accounting for the compositional structure of the data. We utilize centered log-ratio (CLR) and isometric log-ratio (ILR) transformations to preprocess and transform compositional count data, and utilize a latent multi-output additive GP model to enable prediction of future observations. Our experiments demonstrate that MNLVAE outperforms competing method, offering improved prediction performance across different longitudinal microbiome datasets.