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Motivated by the study of random-access protocols for wireless networks, we consider the hard-core model with Metropolis transition probabilities on finite graphs and investigate the asymptotic behavior of the first hitting time between its stable states in the low-temperature regime. In particular, we develop a novel combinatorial method to show how the order-of-magnitude of this first hitting time depends on the grid sizes and on the boundary conditions for various types of grid graphs. Our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low-temperature limit as side-result. In order to derive these results, we extended the model-independent framework for first hitting times known as "pathwise approach'' to allow for more general initial and target states.