Smarr formula for black holes with primary and secondary scalar hair

In this article, we revisit the thermodynamics of black holes endowed with primary and secondary scalar hair in the shift and Z₂ symmetric subclass of beyond Horndeski gravity. Under a specific fine-tuning of the scalar parameter q in terms of the black hole mass, the singular black hole solution with primary scalar hair reduces to the regular Bardeen solution featuring secondary scalar hair. We first demonstrate that the traditional thermodynamic approach fails to yield a consistent Smarr formula for both solutions under consideration. To address this issue, we adopt the approach introduced in Phys. Rev. Lett. 132, 191401 (2024). , and we derive both the first law of black hole thermodynamics and the Smarr formula, offering a consistent thermodynamic description for scalar-hairy black holes. As an additional outcome, our analysis reveals a connection between singular and regular black hole solutions with primary and secondary scalar hair, respectively.