There is good hiking in the Montagne Noir (where it is usually a few degrees cooler than in the valley), as well as longer walks all around the area. Some walks focus on nature, some on history, others simply have nice views.
An interesting place to visit is the Labyrinthe Verte in Nebias. Its a fun walk / short hike with interesting stone formations and somewhat protected during hot summers.
The Aude is a great place for running as well, offering many options.
When it gets hotter, runs along the Canal du Midi or along the Rigole du Canal du Midi (which supplies the water from the lakes in the Montagne Noir to the Canal) are great, as both are mostly tree-covered. It’s always a few degrees cooler in the Montagne Noir than in the valley.