<p>Artificial spin ices are arrays of coupled nanomagnets, which exhibit a variety of fascinating collective behaviour including emergent magnetic monopoles and novel phase transitions. However, they have mainly been confined to two dimensions due to the challenges inherent to their fabrication and characterisation in three dimensions. Exploiting the third dimension offers new degrees of freedom that can lead to curvature-induced topological effects. Here, using numerical simulations, we investigate the magnetic behaviour of a finite three-dimensional spin lattice: the buckyball artificial spin ice, where the spins are located on the edges of a regular buckyball. This frustrated system has a non-trivial topology that results in a rich spectrum of thermal magnetic behaviour, beginning with a crossover from a Paramagnetic sector to a Spin-Ice sector, followed by the formation of an imperfect charge crystal, before spin order is established in three separate steps. The ground state configuration is described by a pair of robust topological magnetic defects that arise because of the finite curved nature of the lattice. Our work reveals the complex thermodynamics of the buckyball artificial spin ice and paves the way to designing unusual magnetic textures in other curved three-dimensional nanomagnetic systems, exploiting the interplay between topology and the dipolar interaction.</p>