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In SU(N) gluodynamics, above the de-confinement temperature, the effectivepotential has minima at non-zero $A_0$-background fields in the two-loopapproximation. Also, it has a minimum at non-zero chromomagnetic backgroundfield, known as 'Savvidy'-vacuum, which shows up on the one-loop level. In thispaper, we join these two approaches. We formulate, at finite temperature, theeffective action, or the free energy, in SU(2) gluodynamics on the two-looplevel, with both, $A_0$ background and magnetic background present at the sametime, which was not done so far. We provide the necessary representations forboth, effective numerical calculation and high-temperature expansions.  The results are represented as a 3D plot of the real part of the effectivepotential. Also, we reproduce for zero either, the $A_0$-background or themagnetic background, the known minima and compare them. The imaginary part is,on the two-loop level, still present. We mention that, as is known fromliterature for the case without $A_0$-background, the imaginary part iscompensated by the ring ('daisy') diagrams. However, our results reveal anunnatural, singular behavior of the effective potential in the region, wherethe imaginary part sets in. Our conclusion is that one has to go beyond thetwo-loop approximation and its ring improved version, in order to investigatethe minimum of the effective action as a function of $A_0$ and chromomagneticfield, and its stability, at least in the approximation of superdaisy diagrams,i.e., the Hartree approximation in the CJT formalism.

Effective potential of gluodynamics in background of Polyakov loop and colormagnetic field