  For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the
distribution of its local time at 0 as the time parameter tends to infinity.
Under the assumption that the L\'evy measure of the inverse local time is
subexponential this distribution behaves asymtotically as a multiple of the
L\'evy measure. Using spectral representations we find the exact value of the
multiple. For this we also need a result on the asymptotic behavior of the
convolution of a subexponential distribution and an arbitrary distribution on
$\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of
the local time allows us to analyze the process derived via a penalization
procedure with the local time. This result generalizes the penalizations
obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.
