We develop further our analysis of the third law of thermodynamics; in particular, we discuss further conditions ensuring the validity of the third law of thermodynamics in its entropic form (N). The introduction in standard homogeneous thermodynamics of the framework in which the absolute temperature T appears as an independent coordinate for the entropy S is followed by the introduction of a more general framework in which Gibbs thermodynamic space, where only extensive independent coordinates appear, is suitably generalized. General properties of S are also discussed. An analysis of the differential conditions which can ensure the validity of (N) follows. Then, we introduce a condition involving the behaviour of generalized heat capacities along curves leaving the surface T = 0 and we show that, under suitable mathematical conditions, it is equivalent to (N). The physical meaning of this condition is also clarified, and amounts to the impossibility for a system to leave a state at T = 0 without heat absorption. Then, we show that a condition of minimum entropy at T = 0 is again equivalent to (N) under suitable conditions. Some notes about (N) when one allows deformation coordinates to be divergent as T → 0+ and about phase coexistence and mixtures also appear.