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Nanoscale ferroelectric topologies such as vortices, anti-vortices, bubble patterns etc. are stabilized in thin films by a delicate balance of both mechanical and electrical boundary conditions. A systematic understanding of the phase stability of bubble domains, particularly when the above factors act simultaneously, remains elusive. Here we present first-principle-based simulations in combination with scanning probe microscopy of ultrathin epitaxial (001) PbZr0.4Ti0.6O3 heterostructures to address this gap. The simulations predict that as-grown labyrinthine domains will transform to bubbles under combinations of reduced film thickness, increased mechanical pressure and/or improved electrical screening. These topological transitions are explained by a common fundamental mechanism. Namely, we argue that, independently of the nature of the driving force, the evolution of the domain morphology allows the system to conserve its original residual depolarization field. Thereby, the latter remains pinned to a value determined by an external or built-in electric bias. To verify our predictions, we then exploit tomographic atomic force microscopy to achieve the concurrent effect of reducing film thickness and increased mechanical stimulus. The results provide a systematic understanding of phase stability and demonstrate controlled manipulation of nanoscale ferroelectric bubble domains.