There exist many non-hyperbolic groups with superlinear divergence. For instance:

• One-ended relatively hyperbolic groups have exponential divergence. (See Sisto's article On metric relative hyperbolicity.)
• For every integer $d \geq 1$, there exists a right-angled Coxeter group whose divergence is polynomial of degree $d$. (See Dani and Thomas' article Divergence in right-angled Coxeter groups.)
• There exist also groups with exotic divergence functions, coming for instance from small cancellation groups or lacunary hyperbolic groups.

There exist many non-hyperbolic groups with superlinear divergence. For instance:

• One-ended relatively hyperbolic groups have exponential divergence. (See Sisto's article On metric relative hyperbolicity.)
• One-ended acylindrically hyperbolic groups have superlinear divergence.
• For every integer $d \geq 1$, there exists a right-angled Coxeter group whose divergence is polynomial of degree $d$. (See Dani and Thomas' article Divergence in right-angled Coxeter groups.)
• There exist also groups with exotic divergence functions, coming for instance from small cancellation groups or lacunary hyperbolic groups.