The Peregrine breather (PB) solution of the nonlinear Schrödinger equation (NLSE) is an important theoretical model for rogue waves in deep water. This study uses computational fluid dynamics (CFD) simulations to investigate the propagation stability and evolution of PB waves over variable water depths. Different k0hshelf values are considered, covering both deep-water k0hshelf >1.363 and shallow-water k0hshelf <1.363 conditions, to examine the effects of topography, forcing duration, and forcing onset. The results show that even when k0hshelf >1.363 (k0hshelf =1.89 in this study), topography affects the breather wave by inducing defocusing and delaying PB focusing, with the delay increasing as k0hshelf decreases. Under deep-water conditions, longer topographic forcing mainly enhances the focusing delay, and the breather recovers its self-focusing characteristics after the water depth increases. In contrast, under shallow-water conditions, prolonged forcing, with the topographic length reaching twice the background wavelength, disrupts the breather structure and separates it into two wave packets. The forcing onset shows little correlation with the delay. This study is the first to reveal topography-induced delayed focusing of PB waves under variable topographic conditions. and demonstrate their robustness over variable topography, providing guidance for rogue-wave prediction and nearshore disaster mitigation.