  In this work we study a more restricted class of quasi-topological gravity
theories where the higher curvature terms have no contribution to the equation
of motion on general static spherically symmetric metric where $g_{tt} g_{rr}
\ne \mathrm{constant}$. We construct such theories up to quintic order in
Riemann tensor and observe an important property of these theories: the higher
order term in the Lagrangian vanishes identically when evaluated on the most
general non-stationary spherically symmetric metric ansatz. This not only
signals the higher terms could only have non-trivial effects when considering
perturbations, but also makes the theories quasi-topological on a much wider
range of metrics. As an example of the holographic effects of such theories, we
consider a general Einstein-scalar theory and calculate it's holographic shear
viscosity.
