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We investigate the thermodynamic properties of the prototype equi-atomic high entropy alloy (HEA) CoCrFeMnNi by using finite-temperature ab initio methods. All relevant free energy contributions are considered for the hcp, fcc, and bcc structures, including electronic, vibrational, and magnetic excitations. We predict the paramagnetic fcc phase to be most stable above room temperature in agreement with experiment. The corresponding thermal expansion and bulk modulus agree likewise well with experimental measurements. A careful analysis of the underlying entropy contributions allows us to identify that the originally postulated dominance of the configurational entropy is questionable. We show that vibrational, electronic, and magnetic entropy contributions must be considered on an equal footing to reliably predict phase stabilities in HEA systems.