  O-minimal geometry generalizes both semialgebraic and subanalytic geometries,
and has been very successful in solving special cases of some problems in
arithmetic geometry, such as Andr\'e-Oort conjecture. Among the many tools
developed in an o-minimal setting are cohomology theories for
abstract-definable continuous manifolds such as singular cohomology, sheaf
cohomology and \v Cech cohomology, which have been used for instance to prove
Pillay's conjecture concerning definably compact groups. In the present paper
we elaborate an o-minimal de Rham cohomology theory for abstract-definable
$\mathcal{C}^\infty$ manifolds in an o-minimal expansion of the real field
which admits smooth cell decomposition and defines the exponential function. We
can specify the o-minimal cohomology groups and attain some properties such as
the existence of Mayer-Vietoris sequence and the invariance under
abstract-definable $\mathcal{C}^\infty$ diffeomorphisms. However, in order to
obtain the invariance of our o-minimal cohomology under abstract-definable
homotopy we must, working in a tame context that defines sufficiently many
primitives, assume the validity of a statement related to Br\"ocker's question.
