  We provide conditions for a compact gradient hyperbolic Ricci and a compact
gradient hyperbolic Yamabe soliton to be trivial, hence, the manifold to be an
Einstein manifold in the first case, and a manifold of constant scalar
curvature, in the second case. In particular, we prove that for a compact
gradient hyperbolic Yamabe soliton of dimension $>2$, if the second Lie
derivative of the metric in the direction of the potential vector field is
trace-free and divergence-free, then the above conclusion is reached.
