  We analyze the role of the symmetry energy slope parameter $L$ on the {\it
r}-mode instability of neutron stars. Our study is performed using both
microscopic and phenomenological approaches of the nuclear equation of state.
The microscopic ones include the Brueckner--Hartree--Fock approximation, the
well known variational equation of state of Akmal, Pandharipande and Ravenhall,
and a parametrization of recent Auxiliary Field Diffusion Monte Carlo
calculations. For the phenomenological approaches, we use several Skyrme forces
and relativisic mean field models. Our results show that the {\it r}-mode
instability region is smaller for those models which give larger values of $L$.
The reason is that both bulk ($\xi$) and shear ($\eta$) viscosities increase
with $L$ and, therefore, the damping of the mode is more efficient for the
models with larger $L$. We show also that the dependence of both viscosities on
$L$ can be described at each density by simple power-laws of the type
$\xi=A_{\xi}L^{B_\xi}$ and $\eta=A_{\eta}L^{B_\eta}$. Using the measured spin
frequency and the estimated core temperature of the pulsar in the low-mass
X-ray binary 4U 1608-52, we conclude that observational data seem to favor
values of $L$ larger than $\sim 50$ MeV if this object is assumed to be outside
the instability region, its radius is in the range $11.5-12$($11.5-13$) km, and
its mass $1.4M_\odot$($2M_\odot$). Outside this range it is not possible to
draw any conclusion on $L$ from this pulsar.
