  The saddle point approximation to the partition functions is an important way
of deriving the thermodynamical properties of black holes. However, there are
certain black hole models and some mathematically analog mechanical models for
which this method can not be applied directly. This is due to the fact that
their action evaluated on a classical solution is not finite and its first
variation does not vanish for all consistent boundary conditions. These
problems can be dealt with by adding a counter-term to the classical action,
which is a solution of the corresponding Hamilton-Jacobi equation. In this work
however, we seek an alternative solution to this problem via the polymer
quantization which is motivated by the loop quantum gravity.
