  We consider the geometric entropy of free nonrelativistic fermions in two
dimensions and show that it is ultraviolet finite for finite fermi energies,
but divergent in the infrared. In terms of the corresponding collective field
theory this is a {\em nonperturbative} effect and is related to the soft
behaviour of the usual thermodynamic entropy at high temperatures. We then show
that thermodynamic entropy of the singlet sector of the one dimensional matrix
model at high temperatures is governed by nonperturbative effects of the
underlying string theory. In the high temperature limit the ``exact''
expression for the entropy is regular but leads to a negative specific heat,
thus implying an instability. We speculate that in a properly defined two
dimensional string theory, the thermodynamic entropy could approach a constant
at high temperatures and lead to a geometric entropy which is finite in the
ultraviolet.
