How do you know what is the fate of the universe in a modified theory of gravity? Usually when you talk about the final stage of the universe you assume that $\Lambda CDM$ is the correct cosmological model and with that you can analyze what could happen for different values of $\Lambda$, my question is, how do you know about the final stage of the  Universe in a different cosmological model? 
I don't have any specified model in mind, you can choose the model you want, I only want to know that could happen in other models.
(I know that $\Lambda CDM$ is the most successful model today)
 A: In Nikodem J. Poplawski's inflationary "cosmology with torsion", a multiverse of temporal iterations (AKA "Local Universes") that's detailed in numerous papers (available free on Arxiv) that he's written between 2009 & 2019 thru use of the relativistic Einstein-Cartan Theory (developed by E-C about 1930, and mathematically more complex than 1915's General Relativity), the gravitational collapse of each astrophysical black hole (formed after the depletion of its nuclear fuel leaves one or another of the largest rotating stars without pressure adequate to support its own mass) results in the propagation (outward from the collapsing star's center) of a causally-separating horizon that separates particles from antiparticles that had formed "virtual pairs" with them:  The in-falling trajectories of the separated fermions (which, like all fermions in ECT, have a tiny spatial extent, as well as spin) are reversed and greatly accelerated outward by contact with the stellar fermions (which, while also spinning, are vastly larger, by an exponent of 32).  
Under Maupertuis' "Principle of Least Action" (from which GR can, reportedly, be derived), these motions are described as resulting from spatial expansion, rather than relative motion. (Although the math involved is opaque to me, I figure this distinction resides in the asymptotically irreversible nature of the causal separation, at least for the new fermions whose materialization had resulted from the aforementioned "tidal effects" of gravity in the causally-separating regions.)  An epoch resembling our own (but on a much smaller scale) occurs while their motion becomes more-or-less inertial (in a region, or L.U., whose shape Poplawski analogizes to a 3-D version of the skin of a basketball), and continues expanding indefinitely, eventually separating them sufficiently to eliminate all resemblance to our current local surroundings, before perhaps forming new pairs with antiparticles from an LU on a spatio / temporal scale like their own, and entering into a reiteration of the process.  
With pi being as varied as it is, it may be quite a while before any iteration actually resembles our own very closely, but the multiverse containing them may continue permanently, and may, in fact, have lacked any "beginning".  (The Borde-Guth-Vilenkin Theorem, which provides logical reasons for believing that an inflationary multiverse might be "geodesically incomplete" to the past, applies only to multiverses that are "on average expanding", so that the sequential reductions in spatial scale that characterize Poplawski's LUs could evade it by an even averaging of expansion against such contractions; also, the "teleparallel" equivalent of geodesics in ECT, that's necessitated by its more sophisticated mathematics, differs from them.)
As discussed in one of Poplawski's Arxiv papers (his 2015 collaboration with Desai, titled "Non-parametric reconstruction of an inflaton potential"), each LU has a local "Big Bang" (or, under some circumstances, a sequence of them), which remains faintly visible as Cosmic Microwave Background radiation.  Concerning other evidence for his cosmology, the interiors of "parented" black holes are, of course, inaccessible and invisible from the "parent", but their exteriors are evident from the elliptical (or nearly circular) orbits of stars whose binary partners they once were. 
As he has been (like Einstein) motivated by the speculations of Mach concerning rotation, Poplawski has considered the most fundamental observational proof of his cosmology, rendering it scientifically verifiable, to be a universal rotation, presumably as evidence for the transmission of torsion between his visualized sequences of spinning spherical bodies on different scales:  Two or three attempts at its verification have been made in recent years, by Cai and others, but their overall results have remained as inconclusive as the attempts to verify the major alternative origin of cosmic inflation (thru a hypothesized scalar field) by the identification of bosons resembling the Higgs (but less massive) have been.  (Re all these remarks of mine, it's generally considered that most stars are in binary pairs, and even more generally considered that virtually all stars rotate, although in many cases their rotation may be only residual, and consequently faint.)     
