  We review thermodynamic properties of modified gravity theories such as
$F(R)$ gravity and $f(T)$ gravity, where $R$ is the scalar curvature and $T$ is
the torsion scalar in teleparallelism. In particular, we explore the
equivalence between the equations of motion for modified gravity theories and
the Clausius relation in thermodynamics. In addition, thermodynamics of the
cosmological apparent horizon is investigated in $f(T)$ gravity. We show both
equilibrium and non-equilibrium descriptions of thermodynamics. It is
demonstrated that the second law of thermodynamics in the universe can be met
when the temperature of the outside of the apparent horizon is equivalent to
that of the inside of it.
