We consider a system of bosonic atoms in an axially symmetric harmonic trap augmented with a two- dimensional repulsive Gaussian optical potential. We find an expression for the grand free energy of the system for configurations ranging from the harmonic trap to the toroidal regime. For large tori we identify an acces- sible regime where the ideal-gas thermodynamics of the system are found to be independent of toroidal radius. This property is a consequence of an invariant extensive volume of the system that we identify analytically in the regime where the toroidal potential is radially harmonic. In considering corrections to the scale-invariant transition temperature, we find that the first-order interaction shift is the dominant effect in the thermodynamic limit and is also scale invariant. We also consider adiabatic loading from the harmonic-to-toroidal trap con- figuration, which we show to have only a small effect on the condensate fraction of the ideal gas, indicating that loading into the scale-invariant regime may be experimentally practical.