We shifted the radial coordinate in such a way that the metric expansion for $q\to \infty$ is of the form $g_{tt}=q^2/l^2+O(1)$, namely there is no linear term in $q$.

Moreover, we will not consider the possible variation of the cosmological constant. For examples of analysis where this is done, refer to [56, 72]

We verified this up to values of charges $Q_0=350$.

This is different than the Kerr-Newman black hole case. The latter configuration is recovered for $\mathrm{\Delta }=0$, therefore the only possible solution to eq. (34) is $\theta =\pi /2$ and $q=0$. In other words, the singularity of the Kerr-Newman solution is located at the equatorial plane and it is a ring. The solution (25) instead has a surface-like singularity, whose radius varies with $\theta$.

In these coordinates the metric expansion for $q\to \infty$ is already of the form $g_{tt}=q^2/l^2+O(1)$.

We traded the dependence in $\mathrm{\Delta }$ for the one in $j$ by means of the relation $J=\frac{a_{1}j}{2{\mathrm{\Xi }}^2}$.

Among the limitations of our ansatz, for instance, a rotating solution with both non vanishing vector fields $A_0$ and $A_1$ has not yet been found for the square root prepotential.

We must add the term imposing the Neumann boundary conditions on $\phi$ as well. The form of this term can be read in [33], Table 2, term $S_{-}$, however this has a vanishing contribution on the on-shell action for these solutions, hence we do not report it here.

Alternative but equivalent renormalization procedures involve the addition of topological invariants such as the Gauss–Bonnet term, see i.e. [73, 74].