  We discuss exact regular compact object solutions in higher dimensional
extensions of General Relativity sourced by a phantom scalar field in arbitrary
$D$ spacetime dimensions ($D>2$), for which a central singularity is absent. We
follow a bottom-up approach, by means of which, by imposing the desired form of
the solution to the metric function, we derive the form of the self-interaction
scalar potential, which in general appears to depend on both the scalar-hair
charge and the black-hole mass. We discuss in this context the validity of the
first law of thermodynamics in such systems. Consistency requires the
independence of the potential of the mass, imposing in this way the dependence
of the mass on the scalar charge of a type that varies with the value of $D$,
and according to the no-hair theorem dressing the regular black hole solution
with secondary hair. In $D=3,4$ we demonstrate that the potential depends on
the ratio of the scalar charge over the mass, and thus considered as a
parameter of the theory. This feature, however, does not characterise
higher-dimensional cases. Calculating the $D-$dimensional Kretschmann scalar we
show that it is finite at the centre point $r=0$ for arbitrary $D$, rendering
the solutions regular. The phantom matter content of the theory is also regular
at $r=0$, hence the radial coordinate of our manifold is defined for $r \ge 0$.
We explicitly discuss the cases of $D=3,4,5,6,10$, and demonstrate that we can
have regular, asymptotically flat, black holes with secondary scalar hair.
