Problems in Black Hole Entropy interpretation

In this work some proposals for black hole entropy interpretation are exposed and investigated. In particular I will firstly consider the so called ``entanglement entropy" interpretation, in the framework of the brick wall model, and the divergence problem arising in the one loop calculations of various thermodynamical quantities, like entropy, internal energy and heat capacity. It is shown that the assumption of equality of entanglement entropy and Bekenstein-Hawking one appears to give inconsistent results. These will be a starting point for a different interpretation of black hole entropy based on peculiar topological structures of manifolds with ``intrinsic" thermodynamical features. It is possible to show an exact relation between black hole gravitational entropy (tree level contribution in path integral approach) and topology of these Euclidean space-times. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for entropy for gravitational instantons are proposed in a form which makes the relation between these self evident. Using this relations I shall propose a generalization of Bekenstein-Hawking entropy in which the former and Euler characteristic are related in the equation: $S=\chi A/8$. The results, quoted above, are more largely exposed in previous works. Finally I'll try to expose some conclusions and hypotheses about possible further development of this research.