  We study thermodynamic and classical instability of AdS black holes in
fourth-order gravity. These include the BTZ black hole in new massive gravity,
Schwarzschild-AdS black hole, and higher-dimensional AdS black holes in
fourth-order gravity. All thermodynamic quantities which are computed using the
Abbot-Deser-Tekin method are used to study thermodynamic instability of AdS
black holes. On the other hand, we investigate the $s$-mode Gregory-Laflamme
instability of the massive graviton propagating around the AdS black holes. We
establish the connection between the thermodynamic instability and the GL
instability of AdS black holes in fourth-order gravity. This may show the
Gubser-Mitra conjecture which holds for AdS black holes found from fourth-order
gravity.
