Nonclosed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics

We develop a covariant differential-form framework to define scalar charges for stationary, asymptotically flat black holes in 4-dimensional Einstein-scalar-Gauss-Bonnet gravity with a general scalar coupling function. Contracting the scalar field equation of motion with the horizon generator k yields a nonclosed-form scalar charge, revealing a bulk contribution encoded in a 3-form, which measures the obstruction to its closedness. In the presence of shift-symmetry, this obstruction vanishes and the 2-form scalar charge satisfies a Gauss law, depending solely on boundary data. Geometrically, this reproduces known topological results in the shift-symmetric limit. This framework allows us to analyze the role of the nonclosed scalar charges in black hole thermodynamics through the Smarr formula for more general couplings and provide a covariant, charge-based interpretation of the spontaneous scalarization mechanism, showing how the behavior of the scalar charge and the bulk term capture the instability of scalar-free black holes and the emergence of scalar hair. Our results offer a unified geometric understanding of the role of scalar charges and a physical interpretation to its nonclosedness in terms of the spontaneous scalarization mechanism.