The covariant variational approach to Yang-Mills theory is further
developed. After reviewing the extension to finite temperature, we briefly
recall the effective action for the Polyakov loop and the critical
properties of the deconfinement phase transition within this approach.
The thermodynamics of pure Yang-Mills theory are studied in detail and
the resulting equation of state is compared to lattice data and other
functional methods. In the confined phase, a small but non-zero pressure
is predicted in contrast to physical expectations; we propose possible
improvements to address this issue. Finally, we discuss the combination
of the variational approach with Dyson-Schwinger techniques in order to
extend the method beyond the Gaussian ansatz. It is shown how to apply
this technique to low order Green's functions in pure Yang-Mills theory
at zero temperature, and the inclusion of fermions by the same method
is briefly layed out.