What is a n-th order temperature?

Although a huge stake of problems in non-equilibrium statistical mechanics is related to thermal reservoirs that within the context of a Langevin description are described by Gaussian noises, there is an important set of problems - which go from diffusive motion in little dense media to molecular motors powered by exergonic chemical reactions such as the hydrolysis of ATP - for which the stochastic nature of the noise in the dynamical equations ought to be everything but Gaussian. As a matter of fact, most of those systems are very well described by taking into consideration Poissonian shot-noise terms for mimicking the interaction system-reservoir. If, to a good extent, it is possible to map athermal linear systems into standard reservoir cases by means of defining a canonical temperature, the existence of non-linearities puts the singular nature of a shot-noise reservoir in the limelight. In that case, specious violations of the principles of thermodynamics can emerge. Aiming to reconcile such natural principles with the analytical properties of non-standard reservoirs, we introduce the concept of temperature of n-th order to characterise the typical scale of energy that is represented by the n-th order cumulant of the noise. With that concept in hand, we also analyse the form of the large deviation function of the energy fluxes and fluctuation relations in systems of this kind.