We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We prove that the cohomology of an abelian regular semisimple Hessenberg variety, with respect to the symmetric group action defined by Tymoczko, is a non-negative combination of tabloid representations. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case. As part of our arguments, we obtain inductive formulas for the Betti numbers of regular Hessenberg varieties.