  Computer simulations of first-order phase transitions using standard toroidal
boundary conditions are generally hampered by exponential slowing down. This is
partly due to interface formation, and partly due to shape transitions. The
latter occur when droplets become large such that they self-interact through
the periodic boundaries. On a spherical simulation topology, however, shape
transitions are absent. By using an appropriate bias function, we expect that
exponential slowing down can be largely eliminated. In this work, these ideas
are applied to the two-dimensional Widom-Rowlinson mixture confined to the
surface of a sphere. Indeed, on the sphere, we find that the number of Monte
Carlo steps needed to sample a first-order phase transition does not increase
exponentially with system size, but rather as a power law $\tau \propto
V^\alpha$, with $\alpha \approx 2.5$, and $V$ the system area. This is
remarkably close to a random walk for which $\alpha$ equals 2. The benefit of
this improved scaling behavior for biased sampling methods, such as the
Wang-Landau algorithm, is investigated in detail.
