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In the context of loop quantum gravity, we construct the phase-space ofisolated horizons with genus greater than 0. Within the loop quantum gravityframework, these horizons are described by genus g surfaces with N puncturesand the dimension of the corresponding phase-space is calculated including thegenus cycles as degrees of freedom. From this, the black hole entropy can becalculated by counting the microstates which correspond to a black hole offixed area. We find that the leading term agrees with the A/4 law and that thesub-leading contribution is modified by the genus cycles.Comment: 22 pages, 9 figures. References updated. Minor changes to match  version to appear in Class. Quant. Gra

Phase space and black-hole entropy of higher genus horizons in loop quantum gravity