  We prove a C-theorem within the framework of two dimensional quantum field
theories at finite temperature. There exists a function C(g) of coupling
constants which is non-increasing along renormalization group trajectories and
non-decreasing along temperature trajectory and stationary only at the fixed
points. The connection between the C-theorem at zero temperature and the
C-theorem at finite temperature is discussed. We also consider the
thermodynamical aspects of the C-theorem. If we define the C-function in an
arbitrary number of dimensions in anology to the two dimensional case, we can
show that its behavior is not universal. The phase transitions destroy the
monotonic properties of the C-function. The proof of the C-theorem is also
presented within the framework of the Kallen-Lehmann spectral representation at
finite temperature.
