We investigate the Glauber dynamics of the generalized (2+1)-dimensional p-SOS model( \(1<p<\infty \) ) under a hard floor constraint. This setting induces entropic repulsion: the integer-valued interface height is forced to remain above the wall and consequently rises to a typical height H(p, L) that depends on both the parameter p and the lattice size L. The phenomenon of entropic repulsion has been extensively studied in a variety of random interface models. In the classical SOS model ( \(p=1\) ), [8, 9] derived an exponential lower bound for the mixing time, demonstrating that the Glauber dynamics mixes only after an exponentially long time in the low-temperature regime (large \(\beta \) , the inverse temperature). However, beyond this case, no rigorous lower bounds were previously known: even for the widely studied Discrete Gaussian model ( \(p=2\) ), the metastable slowdown predicted by the entropic repulsion picture had remained an open problem. On the equilibrium side, [18] obtained sharp large-deviation principles and precise estimates of typical and maximal heights for all \(1\le p<\infty \) , but the dynamical consequences of these results had not been established. Our main contribution is to close this gap by proving that exponentially slow(stretched-exponential) mixing arising from entropic repulsion persists throughout the regime \(1<p<\infty \) . Specifically, we establish an exponential lower bound, showing that the mixing time satisfies \(\tau _{\textrm{mix}}\ge \exp {\left( cL^{1-o(1)}\right) }\) for some \(c>0\) depending on p and \(\beta \) . In addition, we provide a refined metastability analysis, proving that the hitting time of an intermediate level aH(p, L) is at least \(\exp {\left( cL^{a^{d(p)}-o(1)}\right) }\) , where \(0<a<1\) and d(p) is a positive function depending on p. The proof relies on an extension of the classical Peierls-type contour estimates, originally developed for \(p=1\) , to the nonlinear p-SOS setting. Taken together, these results demonstrate that entropic repulsion induces uniformly slow mixing across the entire p-SOS family, thereby extending a phenomenon that had previously been established only for \(p=1\) .
