Spinodoid structures, also called spinodoid metamaterials, are non-periodic cellular structures that mimic spinodal topologies that are observed during diffusion-driven phase separation processes. Computationally efficient to model, spinodoid structures can be fabricated using additive techniques and offer an attractive route to the design of multifunctional structures. In this paper, we investigate the normal incidence sound absorption behavior of four distinct spinodoid topologies: isotropic, cubic, columnar, and lamellar. We fabricate the test samples using the fused filament fabrication process and systematically study the effect of the Gaussian Random Field (GRF) parameters on their underlying open pore network and sound absorption behavior. We employ a watershed segmentation-based image processing approach to correlate their pore and throat radii distributions to the GRF parameters. The normal incidence sound absorption properties are experimentally measured using the two-microphone impedance tube test method. Finally, we use a particle-swarm-based inverse characterization approach to extract the bulk properties necessary to model their acoustical behavior by using the Johnson-Champoux-Allard formulation. Our results show that the open pore network and acoustical properties of spinodoid structures are primarily a function of their relative density and wavenumber. Further, while the absorption behavior of the isotropic, cubic, and columnar spinodoids is similar, the lamellar spinodoids display unique low-frequency sound absorption behavior. Overall, all four spinodoid topologies provide favorable sound absorption characteristics, which may be tuned by varying the GRF parameters. The presented work advances the state-of-the art by establishing the feasibility of using additive manufacturing to enable non-periodic porous structures for that can be tuned to simultaneously provide mechanical stiffness and noise dampening capabilities.