  In this work we study static black holes in the regularized 4D
Einstein-Gauss-Bonnet theory of gravity; a shift-symmetric scalar-tensor theory
that belongs to the Horndeski class. This theory features a simple black hole
solution that can be written in closed form, and which we show is the unique
static, spherically-symmetric and asymptotically-flat black hole vacuum
solution of the theory. We further show that no asymptotically-flat,
time-dependent, spherically-symmetric perturbations to this geometry are
allowed, which suggests that it may be the only spherically-symmetric vacuum
solution that this theory admits (a result analogous to Birkhoff's theorem).
Finally, we consider the thermodynamic properties of these black holes, and
find that their final state after evaporation is a remnant with a size
determined by the coupling constant of the theory. We speculate that remnants
of this kind from primordial black holes could act as dark matter, and we
constrain the parameter space for their formation mass, as well as the coupling
constant of the theory.
