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Hubble constant (H₀) tension and tension in the matter fluctuation amplitude (s₈) are fascinating puzzles in cosmology nowadays. The phantom dynamical dark energy model (PDDE), also known as the little sibling of the big rip, is an abrupt event that can happen in the far future evolution of the universe. A recent analysis of the PDDE model based on cosmic microwave backround radiation data shows that the model is a potential candidate to alleviate these tension problems. In this work, we study the background evolution of the universe within the PDDE model. Analysis based on the SNIa+BAO+OHD data shows that the model successfully explains the late-phase acceleration of the universe. Also, the values of the cosmological parameters computed by the PDDE model are consistent with those obtained by the model for the same dataset. However, most of the phantom dark energy models do not give a stable solution in the asymptotic future. In this regard, we address the PDDE model's dynamical stability and test the validity of the generalized second law (GSL) of thermodynamics. We show that the model is dynamically unstable and violates the GSL. The model does not satisfy the convexity condition, so the universe does not behave as an ordinary macroscopic system within the PDDE model.
Naseeba et al. (Mon,) studied this question.