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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.