Due to our interest in

gauge/gravity duality, we are inclined to analyze gravitational theories that

contain more general interaction compared to Einstein’s gravity, so that using

their parameters we can analyze more general fields on their boundaries.

Lovelock’s gravitational theory provides us with the natural way of extending

Einstein-Hilbert action in D dimensions requiring the graviton’s equations of

motion are of second order in time derivative. In three and four dimensions

Lovelock’s theory is the same as Einstein’s theory of gravity and in higher

dimensions, it is obtained by the addition of Euler characteristics to

Einstein-Hilbert action. According to Lovelock’s theory, the most general

gravitational theory in five dimensions is Gauss-Bonnet gravity, which results

from the addition of the Euler characteristic of a four-dimensional manifold to

Einstein-Hilbert action. Two virtues of Gauss-Bonnet gravity are second order equations

of motion and exact blackhole solutions. Inspiring from these favorable

properties, we can construct a more general gravitational action in five

dimensions, requiring that these properties hold for metrics with spherical

symmetry. This interaction is different from the Euler characteristic of a

six-dimensional manifold which is used to construct third-order Lovelock

theory. The new interaction does not have a topological origin and will depend

on parameters of the problem for geometries that do not have the desirable

symmetries. While the full equations in a general background are fourth-order

in derivatives, it is shown that the linearized equations describing gravitons

propagating in the AdS vacua are precisely the same as second-order equations

of Einstein gravity. Choosing the appropriate sign for the kinetic term,

Anti-de Sitter vacua will be stable and we can find blackhole solutions in

different regions of the parameter space in these vacua. In the next step is to

investigate quasi-topological gravity using Lifshitz metric. In this case,

Gauss-Bonnet and Quasi-topological couplings are not independent. It is found

that despite the different asymptotic behavior of AdS and Lifshitz metrics,

they have the same Wald entropy. Eventually, thermodynamic stability of black

hole solutions can be checked by using the logarithmic plot of time vs.

entropy. It is then observed that a negative Gauss-Bonnet coupling or a

positive quasi-topological coupling would result in instability.