Why did Austria declare war on France in August 1813?

Austria declared war on France in August 1813 primarily because Klemens von Metternich failed to secure a diplomatic compromise with Napoleon I.1

Following Napoleon's defeat in Russia in 1812, Metternich sought to establish a stable balance of power in Europe rather than simply restoring the pre-1789 order. He aimed to create a system where neither a fully victorious France nor a fully victorious Russia could dominate the continent, hoping to position the Habsburg monarchy and Prussia as essential counterweights between them.1

To achieve this, Metternich attempted to act as an armed mediator throughout the first half of 1813, demanding concessions from Napoleon such as the return of Illyria, the partition of the duchy of Warsaw, and the dissolution of the Confederation of the Rhine. When Napoleon refused to agree to these terms, Austria formally declared war.21