  In the original holographic dark energy (HDE) model, the dark energy density
is proposed to be $\rho_{de} = 3c^2M^2_{pl}L^{-2}$, with $c$ is a dimensionless
constant characterizing the properties of the HDE. In this work, we propose the
generalized holographic dark energy (GHDE) model by considering the parameter
$c$ as a redshift-dependent function $c(z)$. We derive all the physical
quantities of the GHDE model analytically, and fit the $c(z)$ by trying four
kinds of parametrizations. The cosmological constraints of the $c(z)$ are
obtained from the joint analysis of the present SNLS3+BAO+CMB+$H_0$ data. We
find that, compared with the original HDE model, the GHDE models can provide a
better fit to the data. For example, the GHDE model with JBP-type $c(z)$ can
reduce the $\chi^2_{min}$ of the HDE model by 2.16. We also find that, unlike
the original HDE model with a phantom-like behavior in the future, the GHDE
models can present many more different possibilities, i.e., it allows the GHDE
in the future to be either quintessence like, cosmological constant like, or
phantom like, depending on the forms of $c(z)$.
