We present a new method for the evaluation of the perturbative expansion of the QCD pressure which is valid at all values of the temperature and quark chemical potentials in the deconfined phase and which we work out up to and including order $g^4$ accuracy. Our calculation is manifestly four-dimensional and purely diagrammatic—and thus independent of any effective theory descriptions of high temperature or high density QCD. In various limits, we recover the known results of dimensional reduction and the hard dense/thermal loop (HDL/HTL) resummation schemes, as well as the equation of state of zero-temperature quark matter, thereby verifying their respective validity. To demonstrate the overlap of the various regimes, we furthermore show how the predictions of dimensional reduction and HDL resummed perturbation theory agree in the regime $T\sim \sqrt{g}\mu$. At parametrically smaller temperatures $T\sim g\mu$, we find that the dimensional reduction result agrees well with those of the nonstatic resummations down to the remarkably low value $T\approx 0.2m_{\mathrm{D}}$, where $m_{\mathrm{D}}$ is the Debye mass at $T=0$. Beyond this, we see that only the latter methods connect smoothly to the $T=0$ result of Freedman and McLerran, to which the leading small-$T$ corrections are given by the so-called non-Fermi-liquid terms, first obtained through HDL resummations. Finally, we outline the extension of our method to the next order, where it would include terms for the low-temperature entropy and specific heats that are unknown at present.

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