We present a new family of regular black holes (RBH) in Pure Lovelock gravity, where the energy density is determined by the gravitational vacuum tension, which varies for each value of n in each Lovelock case. Speculatively, our model may capture quantum effects through gravitational tension. In this way, a hypothetical analogy is drawn between the pair production ratio in the Schwinger effect and our energy density. A notable feature of our model is that the regular solution closely resembles the vacuum solution before reaching the event horizon. For odd n, the transverse geometry is spherical, with phase transitions occurring during evaporation, and the final state of this process is a remnant. For even n, the transverse geometry is non trivial and corresponds to a hyperboloid. In the case of d = 2n+1 with even n, we find an RBH without a dS core and no inner horizon (whose presence has been recently debated in the literature due to the question of whether its presence is unstable or not), and no phase transitions. For d > 2n + 1 with even n, the RBH possesses both an event horizon and a cosmological horizon, also with no inner horizon present. The existence of the cosmological horizon arises without the usual requirement of a positive cosmological constant. From both numerical and analytical analysis, we deduce that as the event horizon expands and the cosmological horizon contracts, thermodynamic equilibrium is achieved in a remnant when the two horizons coincide.