  The evolution of a driven quantum system is said to be adiabatic whenever the
state of the system stays close to an instantaneous eigenstate of its
time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures
that such pure state adiabaticity can be maintained with arbitrary accuracy,
provided one chooses a small enough driving rate. Here, we extend the notion of
quantum adiabaticity to closed quantum systems initially prepared at finite
temperature. In this case adiabaticity implies that the (mixed) state of the
system stays close to a quasi-Gibbs state diagonal in the basis of the
instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition
for the finite temperature adiabaticity. Remarkably, it implies that the finite
temperature adiabaticity can be more robust than the pure state adiabaticity,
particularly in many-body systems. We present an example of a many-body system
where, in the thermodynamic limit, the finite temperature adiabaticity is
maintained, while the pure state adiabaticity breaks down.
