From the statistical mechanical point of view, area-preserving maps have great potential and importance. These maps exhibit chaotic and regular behavior separately or together in the available phase space as the control parameter changes. Several works on these maps, e.g., the standard map and the web map, have shown that ergodicity breakdown causes the statistical mechanical framework that describes the dynamics of the system to change. In this paper, for a novel generalization of the standard map, which we define by generalizing the periodic function used in its definition, we verify that a q-Gaussian with \(q\simeq 1.935\) for the probability distribution of sum of the iterates of the system with initial conditions chosen from the nonergodic stability islands is robust. We also show that the probability distributions become more complicated and unexpected limiting behavior occurs for some parameter regimes.