Why is $\frac{d^2U}{dx^2}$ constant close to the equilibrium point?

It's not that $\frac{d^2U}{dx^2}$ is constant close to the equilibrium point. The idea is that close to the equilibrium point, the Taylor expansion $U(x) = U_0 + x\left(\frac{dU}{dx}\right)_{0} + \frac{x^2}{2!}\left(\frac{d^2U}{dx^2}\right)_0 + ...$, holds. In this expression, each derivative is evaluated at $x_0$. So the value of $\frac{d^2U}{dx^2}$ at the equilibrium point is what matters, if you are trying to understand behavior near the equilibrium.