  Recently, scalarization of Schwarzschild black hole are extensively studied.
In this work, we explore the scalarization of Taub-NUT black hole. The theory
we consider is the extended scalar-tensor-Gauss-Bonnet theory, which admits
Ricci-flat Taub-NUT black hole as a solution. An analysis of probe scalar field
is carried out to identify the mass parameter and NUT parameter (m,n) where the
hairy black holes start to emerge. Then, we use shooting method to construct
the scalarized Taub-NUT black hole numerically. Being different from the
Schwarzschild case, there exists two branches of new hairy black holes which
are smoothly connected to each other. We calculate the entropy of scalarized
black holes and compare it with the entropy of scalar-free Taub-NUT black
holes, it turns out that the entropy of the new hairy black holes are larger
than that of scalar-free black holes. A novel phenomena emerges in this system
that the entropy of the black holes at the bifurcation point is constant for
positive mass parameter. We then conjecture a maximal entropy bound for all the
scalarized black hole whose mass parameter at the bifurcation point is greater
than zero.
