Testing the dynamical stability and validity of generalized second law within the phantom dynamical dark energy model

Hubble constant($H_0$) tension and tension in the matter fluctuation amplitude ($s_8$) are fascinating puzzles in cosmology nowadays. Phantom dynamical dark energy model (PDDE), also known as little sibling of the big rip is an abrupt event that can happen in the far future evolution of the universe. Recent analysis of PDDE model based on CMBR 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 is successful in explaining the late phase acceleration of the universe. Also, the values of the cosmological parameters predicted by PDDE model are consistent with the values predicted by the $\Lambda$CDM model. However, most of the phanton dark energy models doesn't give stable solution in the asymptotic future. In this regard, we address the dynamical stability of the PDDE model and also 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 doesn't satisfy the convexity condition and hence the universe doesn't behave like an ordinary macroscopic system within the PDDE model.