Using the Sharma-Mittal entropy, we study some properties of the Schwarzschild and Schwarzschild-de Sitter black holes. The results are compared with those obtained by attributing the Bekenstein entropy bound to the mentioned black holes. Our main results show that while the Schwarzschild black hole is always stable in the micro-canonical ensemble, it can be stable in the canonical ensemble if its mass is bigger than the mass of the coldest Schwarzschild black hole. A semi-classical analysis has also been used to find an approximate relation between the entropy free parameters. Throughout the paper, we use units $c=G=\hbar=k_B=1$, where $k_B$ denotes the Boltzmann constant.
