  We derive a mass formula and a mass variation law for asymptotically flat,
stationary spacetimes, invariant under two commuting rotational symmetries, in
a general five dimensional theory of gravity coupled to an arbitrary set of
Maxwell fields and uncharged scalar fields. If the spacetime is everywhere
regular, these mass formulas reduce to a sum of magnetic flux terms defined on
its non-trivial 2-cycles. If there is a black hole, we obtain a mass variation
law more general than previously obtained, which also has contributions from
the 2-cycles exterior to the black hole. This can be interpreted as the first
law of black hole mechanics in a background soliton containing bubbles.
