  We revisit the first law of black hole thermodynamics in 4-dimensional
theories containing scalar and Abelian vector fields coupled to gravity using
Wald's formalism and a new definition of scalar charge as an integral over a
2-surface which satisfies a Gauss law in stationary black-hole spacetimes. We
focus on ungauged supergravity-inspired theories with symmetric sigma models
whose symmetries generate electric-magnetic dualities leaving invariant their
equations of motion. Our manifestly duality-invariant form of the first law is
compatible with the one obtained by of Gibbons, Kallosh and Kol. We also obtain
the general expression for the scalar charges of a stationary black hole in
terms of the other physical parameters of the solution and the position of the
horizon, generalizing the expression obtained by Pacilio for dilaton black
oles.
