  We introduce the notion of asymptotic Grothendieck groups for abelian and
triangulated categories that are both AB4 and AB4*. We study when the
asymptotic Grothendieck group of the heart of a triangulated category with a
t-structure is isomorphic to the asymptotic Grothendieck group of the
triangulated category itself. We also explain a connexion with the notion of
topological Grothendieck group from Achar-Stroppel. Finally, we compute the
asymptotic Grothendieck group of some positive (dg-)algebras having a cone
bounded, locally finite dimension.
