  Dark matter models with light mediators featuring sizable interactions among
dark particles enjoy an increasing attention in the model building community
due to the elegance with which they can potentially explain the scaling
relations governing galactic halos and clusters of galaxies. In the present
work we continue our study of such models using non-relativistic and potential
non-relativistic effective field theories (NREFTs and pNREFTs) and explore the
properties of a Yukawa-type model with scalar and pseudoscalar interactions
between a low-energetic scalar mediator and heavy dark matter fermions. In
particular, we make first steps towards the formulation of such theories at
finite temperature by providing the thermal bound-state formation rate and the
thermal break-up of bound states from the self-energies of the dark-pair
fields, that interact with the thermal environment. We estimate numerically
bound-state effects on the dark matter energy density, that provide up to a
$35\%$ correction depending on the relative size of the model couplings.
