In 1899, he was appointed professor of modern German language and literature at the University of Leipzig, succeeding Rudolf Hildebrand.
During his time in Leipzig, he received calls to prestigious chairs, for example as successor to his mentor and friend Erich Schmidt at the Humboldt-Universität zu Berlin or at the University of Vienna, which he declined, however.
In addition to his achievements in the field of literary history, Köster was one of the first scholars to strive for methodically founded scientific theatre research.
This was acquired by the Deutsches Theatermuseum after his death, but fell victim for the most part to a bombing raid in 1944.
At his request, his extensive library was sold at auction by Walter de Gruyter & Co. 26/27 January 1925.