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We discuss the dependence of pure Yang-Mills equation of state on the choiceof gauge algebra. In the confined phase, we generalize to an arbitrary simplegauge algebra Meyer's proposal of modelling the Yang-Mills matter by an idealglueball gas in which the high-lying glueball spectrum is approximated by aHagedorn spectrum of closed-bosonic-string type. Such a formalism is undefinedabove the Hagedorn temperature, corresponding to the phase transition toward adeconfined state of matter in which gluons are the relevant degrees of freedom.Under the assumption that the adjoint string tension and the typical energyscale of the running coupling are gauge-algebra independent, we discuss abouthow the behavior of thermodynamical quantities such as the trace anomaly shoulddepend on the gauge algebra in both the confined and deconfined phase. Theobtained results compare favourably with recent and accurate lattice data inthe $\mathfrak{su}(3)$ case and support the idea that the more the gaugealgebra has generators, the more the phase transition is of first-order type.

Comments on Yang-Mills thermodynamics, the Hagedorn spectrum and the gluon gas