  Building on works of Boulanger and Goto, we show that Goto's scalar curvature
is the moment map for an action of generalized Hamiltonian automorphisms of the
associated Courant algebroid, constrained by the choice of an adapted volume
form. We derive an explicit formula for Goto's scalar curvature, and show that
it is constant for generalized K\"ahler-Ricci solitons. Restricting to the
generically symplectic type case, we realize the generalized K\"ahler class as
the complexified orbit of the Hamiltonian action above. This leads to a natural
extension of Mabuchi's metric and $K$-energy, implying a conditional uniqueness
result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz
obstruction and a Futaki invariant.
