  This paper is a continuation of the paper \emph{Low regularity Cauchy problem
for the fifth-order modified KdV equations on $\mathbb{T}$}. In this paper, we
consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as
following: \begin{equation*} \begin{cases} \partial_t u - \partial_x^5 u +
30u^2\partial_x u + 20 u\partial_x u \partial_x^3u + 10u \partial_x^3 u = 0,
\hspace{1em} (t,x) \in \mathbb{R} \times \mathbb{T}, u(0,x) = u_0(x) \in
H^s(\mathbb{T}) \end{cases}. \end{equation*}
  We prove the local well-posedness of the fifth-order KdV equation for low
regularity Sobolev initial data via the energy method. This paper follows
almost same idea and argument as in the paper \emph{Low regularity Cauchy
problem for the fifth-order modified KdV equations on $\mathbb{T}$}. Precisely,
we use some conservation laws of the KdV Hamiltonians to observe the direction
which the nonlinear solution evolves to. Besides, it is essential to use the
short time $X^{s,b}$ spaces to control the nonlinear terms due to \emph{high
$\times$ low $\Rightarrow$ high} interaction component in the non-resonant
nonlinear term. We also use the localized version of the modified energy in
order to obtain the energy estimate.
  As an immediate result from a conservation law in the scaling sub-critical
problem, we have the global well-posedness result in the energy space $H^2$.
