  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 accessible 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 configuration, 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.
