= Entropy density at zero chemical potential
{title2=$s=(\rho+P)/T$}
For an extensive equilibrium system, the <thermodynamic Euler relation> gives $E=TS-PV+\mu N$. At zero <chemical potential>, divide by $V$ to obtain $s=(\rho+P)/T$. Equivalently, $dP=(\rho+P)dT/T$ makes
$$
d\left(\frac{(\rho+P)V}{T}\right)=\frac{d(\rho V)+P\,dV}{T}.
$$
The <first law of thermodynamics> identifies this as $dS$; the <thermodynamic Euler relation> fixes the entropy normalization. For adiabatic expansion satisfying the <cosmological perfect-fluid continuity equation>, the numerator on the right vanishes, proving <cosmological entropy conservation>.
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