is therefore easy to check that the S-deformation is licit, $S(r^*=+\infty)=S(r^*=-\infty)=0$. So, we can directly use our conditions of stability which shows that outside the horizon, in the range $M<r<2M$, we have ${\cal F},~{\cal G}$ and ${\cal H}$ negative.

In conclusion the BBMB solution is unstable under odd gravitational perturbations.

Extended BBMB solution
----------------------

In [@Martinez:2002ru], the authors extended the solution by including a cosmological constant (MTZ solution) $$\begin{aligned}
ds^2 &=-\Bigl[\Bigl(1-\frac{M}{r}\Bigr)^2-\frac{\Lambda}{3}r^2\Bigr] dt^2+\frac{dr^2}{\Bigl(1-\frac{M}{r}\Bigr)^2-\frac{\Lambda}{3}r^2}\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+r^2d\Omega^2\,,\\
\phi &=\frac{\sqrt{6}M}{r-M}\,.\end{aligned}$$ For that, they had to add in the action, a quartic interaction term along with a cosmological constant $$\begin{aligned}
S=\int{\rm d}^4x\sqrt{-g}\Bigl[\frac{R}{2}-\frac{1}{2}(\partial\phi)^2-\frac{R}{12}\phi^2-\Lambda+\frac{\Lambda}{36}\phi^4\Bigr]\,.\end{aligned}$$ This solution is studied between the event horizon $r_\text{min}$ and the cosmological horizon $r_\text{max}$, given by $$\begin{aligned}
r_\text{min} &=\frac{l}{2}\Bigl(1-\sqrt{1-\frac{4M}{l}}\Bigr)\,,\\
r_\text{max} &=\frac{l}{2}\Bigl(1+\sqrt{1-\frac{4M}{l}}\Bigr)\,,\end{aligned}$$ where $l=\sqrt{3/\Lambda}>0$ and hence, the existence of the horizon imposes the condition $0<4M<l$. For this class of theories, we find that the S-deformation is licit, $S(r^*=+\infty)=S(r^*=-\infty)=0$, and the stability condition is given by the positivity of $${\cal F}={\cal G}={\cal H}=1-\frac{\phi^2}{6}=\frac{r(2-2M)}{(r-M)^2}\,.$$ It is again easy to check that the odd perturbations are unstable because ${\cal F},~{\cal G}$ and ${\cal H}$ are negative in the range $r_\text{min}<r<2M$. In fact in the range of existence of the black hole solution, $0<4M<l$, we always have $r_\text{min}<2M$.

No 