We study cosmological perturbations in mimetic matter scenario with a general higher derivative function. We calculate the quadratic action and show that both the kinetic term and the gradient term have the wrong sings. We perform the analysis in both comoving and Newtonian gauges and confirm that the Hamiltonians and the associated instabilities are consistent with each other in both gauges. The existence of instabilities is independent of the specific form of higher derivative function which generates gradients for mimetic field perturbations. It is verified that the ghost instability in mimetic perturbations is not associated with the higher derivative instabilities such as the Ostrogradsky ghost. * firouz@ipm.ir † gorji@ipm.ir ‡ shosseini@shahroodut.ac.ir, shossein@ipm.ir

1 We use the metric signature convention (−, +, +, +). 2 We work in natural unit in which M Pl = 1 where M Pl is the reduced Planck mass.

We explore a formulation of the thermodynamic geometry of black holes and prove that the divergent points of the specific heat correspond exactly to the singularities of the thermodynamic curvature. We investigate this correspondence for different types of black holes. This formulation can also be applied to an arbitrary thermodynamic system.

Following a methodology similar to [1], we derive a holographic complexity for two dimensional holographic superconductors (gauge/string superconductors) with backreactions. Applying a perturbation method proposed by Kanno in Ref. [2], we study behaviors of the complexity for a dual quantum system near critical points. We show that when a system moves from the normal phase (T > Tc) to the superconductor phase (T < Tc), the holographic complexity will be divergent.