Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as β, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the <i>q</i>-dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell-Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ/3, where Λ represents the cosmological constant within the ΛCDM standard model for cosmology. For a simple gas composed of particles with proper mass <i>m</i>, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter <i>q</i> exhibits a dependence on the cosmological constant given by q=1+ℓcΛ/n, where ℓc=ℏ/mc is the Compton length of the particle and n is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters. Abstract Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as β , is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q -dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell–Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ / 3 , where Λ represents the cosmological constant within the Λ CDM standard model for cosmology. For a simple gas composed of particles with proper mass m , and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by q = 1 + ℓ c Λ / n , where ℓ c = ℏ / m c is the Compton length of the particle and n is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters. [ 1 ] about the relation between entropy invariance and relativistic variance of temperature (translated from French): It is well known that entropy, alongside the space-time interval, electric charge, and mechanical action, is one of the fundamental “invariants” of the theory of relativity. To convince oneself of this, it is enough to recall that, according to Boltzmann, the entropy of a macroscopic state is proportional to the logarithm of the number of microstates that realize that state. To strengthen this reasoning, one can argue that, on the one hand, the definition of entropy involves a integer number of microstates, and, on the other hand, the transformation of entropy during a Galilean reference frame change must be expressed as a continuous function of the relative velocity of the reference frames. Consequently, this continuous function is necessarily constant and equal to unity, which means that entropy is constant. Let us now give more insights about what “relativistic thermodynamics” could be. In relativistic thermodynamics (i.e., in accordance with special relativity), there exist three points of view [ 2 ], distinguished from the way heat Δ Q and temperature T transform under a Lorentz boost from frame R 0 (e.g., laboratory) to comoving frame R with velocity v = v n ^ relative to R 0 and Lorentz factor (1) γ ( v ) = 1 1 − v 2 / c 2 . ( a ) The covariant viewpoint (Einstein [ 3 ], Planck [ 4 ], de Broglie [ 1 ] …), (2) Δ Q = Δ Q 0 γ − 1 , T = T 0 γ − 1 . ( b ) The anti-covariant one (Ott [ 5 ], Arzelies [ 6 ], …), (3) Δ Q = Δ Q 0 γ , T = T 0 γ . ( c ) The invariant one, “nothing changes” (Landsberg [ 7 , 8 ], …), (4) Δ Q = Δ Q 0 , T = T 0 . Also note that, for some authors (Landsberg [ 9 ], Sewell [ 10 ], …), “there is no meaningful law of temperature under boosts”. Nevertheless, more recent approaches (e.g., Ref. [ 11 ]) show that there is a covariant relativistic thermodynamics with proper absolute temperature in full agreement with relativistic hydrodynamics. In this paper, we adopt the viewpoint in Section 1 and review de Broglie’s arguments in Section 2 . In Section 3 , we remind you of the construction of the so-called Maxwell–Jüttner distribution presented by Synge in Ref. [ 12 ]. In Section 4 , we then present the de Sitter space-time, its geometric description as a hyperboloid embedded in the 1 + 4 Minkowski space-time, and give some insights of the fully covariant quantum field theory of free scalar massive elementary systems propagating on this manifold. In Section 5 , we then develop our arguments in favor of a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters. A few comments end our paper in Section 6 . 2. Inverse Temperature Four-Vector The found distribution ( 30 ) on the Minkowskian mass shell for a simple gas consisting of particles of proper mass m leads us to introduce the relativistic thermodynamic, future directed, time-like four-coldness vector β _ , as the four-version of the reciprocal of the thermodynamic temperature (see also Ref. [ 2 ]): (33) c u _ k B T a ≡ β _ = ( β 0 = β 0 > 0 , β i = − β i ) = ( β 0 , β ) , with absolute coldness as relativistic invariant, (34) β _ · β _ = c k B T a ≡ β a .