  Minimal $D=5$ supergravity admits asymptotically globally AdS$_5$
gravitational solitons (strictly stationary, geodesically complete spacetimes
with positive mass). We show that, like asymptotically flat gravitational
solitons, these solutions satisfy mass and mass variation formulas analogous to
those satisfied by AdS black holes. A thermodynamic volume associated to the
non-trivial topology of the spacetime plays an important role in this
construction. We then consider these solitons within the holographic
``complexity equals action'' and ``complexity equals volume'' conjectures as
simple examples of spacetimes with nontrivial rotation and topology. We find
distinct behaviours for the volume and action, with the counterterm for null
boundaries playing a significant role in the latter case. For large solitons we
find that both proposals yield a complexity of formation proportional to a
power of the thermodynamic volume, $V^{3/4}$. In fact, up to numerical
prefactors, the result coincides with the analogous one for large black holes.
