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The Euclidean action and entropy are computed in string-generated gravity models with quadratic curvatures, and used to argue that a negative mass extremal metric is the background for hyperbolic (k=-1) black hole spacetimes, k being the curvature constant of the event horizon. The entropy associated with a black hole is always positive for k=0, 1 family. The positivity of energy condition also ensures that the k=-1 (extremal) entropy is non-negative.
Ishwaree P. Neupane (Thu,) studied this question.
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