The sample scale-free Gini index is known to be a powerful test of exponentiality against a broad class of alternatives. To understand better the efficiency properties of this test we calculate its Bahadur efficiency for most commonly used parametric alternatives to the exponential distribution. Using variational arguments and the Bahadur-Raghavachari inequality for exact slopes we find the conditions of local Bahadur optimality of the Gini test. It turns out that this property surprisingly holds for a family of alternative distributions including the well-known GompertzMakeham distribution.