  We study the generalized second law (GSL) of thermodynamics in $f(T)$
cosmology. We consider the universe as a closed bounded system filled with $n$
component fluids in the thermal equilibrium with the cosmological boundary. We
use two different cosmic horizons: the future event horizon and the apparent
horizon. We show the conditions under which the GSL will be valid in specific
scenarios of the quintessence and the phantom energy dominated eras. Further we
associate two different entropies with the cosmological horizons: with a
logarithmic correction term and a power-law correction term. We also find the
conditions for the GSL to be satisfied or violated by imposing constraints on
model parameters.
