Using space-periodic photonic structures has been a major strategy for the manipulation of spontaneous emission during the past several decades. Recent interest in time-varying photonics, on the other hand, needs an extension of the theory of spontaneous emission to account for its temporal-variation-induced alteration. The photonic temporal crystal (or photonic Floquet medium), whose optical properties are controlled in a time-periodic pattern, is the temporal counterpart of spatially periodic photonic crystals. Due to the coupling between positive- (negative-) frequency bands and replicated negative- (positive-) frequency bands, the Floquet eigenstates for the photonic temporal crystal are non-orthogonal to each other. Non-Hermiticity implies that the relevant phenomena observed in the photonic temporal crystal must be understood within a non-Hermitian framework. Here, we demonstrate how the Petermann factor (PF), in tandem with the non-Hermitian analysis, results in the finite non-zero spontaneous emission rate (SER) at the momentum gap (MG) frequency. In addition, we calculated the spontaneous absorption rate originating from the gain mechanism due to the modulation of the optical properties of the photonic temporal crystals.