  In this paper we examine the behavior in temperature of the free energy on
quantum systems in an arbitrary number of dimensions. We define from the free
energy a function $C$ of the coupling constants and the temperature, which in
the regimes where quantum fluctuations dominate, is a monotonically increasing
function of the temperature. We show that at very low temperatures the system
is controlled by the zero-temperature infrared stable fixed point while at
intermediate temperatures the behavior is that of the unstable fixed point. The
$C$ function displays this crossover explicitly. This behavior is reminiscent
of Zamolodchikov's $C$-theorem of field theories in 1+1 dimensions. Our results
are obtained through a thermodynamic renormalization group approach. We find
restrictions on the behavior of the entropy of the system for a
$C$-theorem-type behavior to hold. We illustrate our ideas in the context of a
free massive scalar field theory, the one-dimensional quantum Ising Model and
the quantum Non-linear Sigma Model in two space dimensions. In regimes in which
the classical fluctuations are important the monotonic behavior is absent.
