We investigate how a negative cosmological constant affects the Gauss-Bonnet (GB) scalarization in the Einstein-Maxwell-scalar-Gauss-Bonnet theory with a scalar coupling constant $\eta$ to GB term. We focus on the instability of Reissner-Nordström-AdS (RN-AdS) black holes under a scalar perturbation governed by an effective mass ${\mu }_{\mathrm{eff}}^2$ sourced by the GB term. Unlike the asymptotically flat spacetime case, the onset of scalarization is not merely determined by ${\mu }_{\mathrm{eff}}^2<0$, but it is constrained by the Breitenlohner-Freedman (BF) bound. In case that the BF bound is violated ($\eta >2.25$ with $\mathrm{\Lambda }=-0.5$), one may find AdS-tachyonic instability. We find that for $0<\eta <2.25$, the ${\mathrm{GB}}^{+}$ scalarization may be performed through spontaneous scalarization, while for $\eta <0$ the ${\mathrm{GB}}^{-}$ scalarization is found to give the single branch of scalarized AdS black holes. For the ${\mathrm{GB}}^{+}$ scalarization in ${\eta }_{\mathrm{th}}\le \eta <2.25$ with ${\eta }_{\mathrm{th}}$ threshold instability, we obtain the single branch ($n=0$ fundamental branch) of scalarized AdS black holes, in contrast to the infinite branches in asymptotically flat spacetime. A bulk fixed-charge thermodynamic analysis is performed thoroughly for ${\mathrm{GB}}^{\pm }$ scalarizations.

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• Alternative gravity theories
• Quantum aspects of black holes