  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.
