  We prove three formulas for computing topological pressure of $C^1$-generic
conservative diffeomorphism and show the continuity of topological pressure
with respect to these diffeomorphisms. We prove for these generic
diffeomorphisms that there is no equilibrium states with positive measure
theoretic entropy. In particular, for hyperbolic potentials, there is no
equilibrium states. For $C^1$ generic conservative diffeomorphism on compact
surfaces with no dominated splitting and $\phi_m(x):=-\frac{1}{m}\log \Vert D_x
f^m\Vert, m \in \mathbb{N}$, we show that there exist equilibrium states with
zero entropy and there exists a transition point $t_0$ for the family $\lbrace
t \phi_m(x)\rbrace_{t\geq 0}$, such that there is no equilibrium states for $ t
\in [0, t_0)$ and there is an equilibrium state for $t \in [t_0,+\infty)$.
