•A Hamiltonian scheme is proposed for the Higgs boson equation for the first time.•The model is a generalization of the Higgs boson equation in the de Sitter space-time.•Generalized potential and fractional spatial diffusion are considered.•The scheme is mathematically analyzed both variationally and numerically.•Also, a simpler linear scheme is analyzed, and a Matlab implementation is proposed. The present work is the first paper in the literature to report on a Hamiltonian discretization of the (fractional) Higgs boson equation in the de Sitter space-time, and its theoretical analysis. More precisely, we design herein a numerically efficient finite-difference Hamiltonian technique for the solution of a fractional extension of the Higgs boson equation in the de Sitter space-time. The model under investigation is a multidimensional equation with generalized potential and Riesz space-fractional derivatives of orders in (1,2. An energy integral for the model is readily available, and we propose a nonlinear, implicit and consistent numerical technique based on fractional-order centered differences, with similar Hamiltonian properties in the discrete scenario. A fractional energy approach is used then to prove the properties of stability and convergence of the technique. For simulation purposes, we consider both the classical and the fractional Higgs real-valued scalar fields in the de Sitter space-time, and find results qualitatively similar to those available in the literature. For the sake of convenience, we provide the Matlab code of an alternative linear discretization of the method presented in this work. This linear implicit approach is thoroughly analyzed also.