In this paper, we shall consider $f\left(R\right)$ gravity and its cosmological implications, when an extra matter term generated by thermal effects is added by hand in the Lagrangian. We formulate the equations of motion of the theory as a dynamical system, that can be treated as an autonomous one only for specific solutions for the Hubble rate, which are of cosmological interest. Particularly, we focus our analysis on subspaces of the total phase space, corresponding to (quasi-)de Sitter accelerating expansion, matter-dominated and radiation-dominated solutions. In all the aforementioned cases, the dynamical system is an autonomous dynamical system. With regard to the thermal term effects, these are expected to significantly affect the evolution near a Big Rip singularity, and we also consider this case in terms of the corresponding dynamical system, in which case the system is non-autonomous, and we attempt to extract analytical and numerical solutions that can assess the specific cases. This course is taken twice: the first for the vacuum theory and the second when two perfect fluids (dust and radiation) are included as matter sources in the field equations. In both cases, we reach similar conclusions. The results of this theory do not differ significantly from the results of the pure $f\left(R\right)$ in the de Sitter and quasi-de Sitter phases, as the same fixed points are attained, so for sure the late-time era de Sitter is not affected. However, in the matter-dominated and radiation-dominated phases, the fixed points attained are affected by the presence of the thermal term, so surely the thermal effects would destroy the matter and radiation domination eras. However, with regard to the Big Rip case, several instabilities are found in the dynamical system, since the initial conditions dramatically affect the behavior of the dynamical system near the starting point of the $e$-foldings number evolution. Finally, we consider the effects of the thermal term on the late-time evolution of the Universe, by examining several statefinder quantities. Our findings indicate that these are reportable only under extreme fine tuning of the multiplicative coefficient of the thermal term.

在本文中，我们将考虑 $F（\mathrm{[R}）$当在拉格朗日中手工添加由热效应产生的额外物质项时，重力及其宇宙学含义。我们将理论的运动方程式表述为一个动力学系统，该动力学方程式仅对于哈勃速率的特定解决方案可以视为自治系统，而哈勃速率具有宇宙学意义。特别是，我们将分析重点放在总相空间的子空间上，这些子空间对应于（准）de Sitter加速膨胀，物质占优和辐射占优的解决方案。在所有上述情况下，动力系统是自主动力系统。关于热项效应，预计这些效应会显着影响Big Rip奇点附近的演化，并且我们还将根据相应的动力学系统来考虑这种情况，在这种情况下，系统是非自治的，并且我们尝试提取可以评估特定案例的分析和数值解决方案。该课程进行了两次：第一阶段是关于真空理论，第二阶段是在场方程中包含两种理想流体（粉尘和辐射）作为物质源。在这两种情况下，我们得出相似的结论。该理论的结果与纯理论的结果没有显着差异$F（\mathrm{[R}）$在de Sitter和准de Sitter阶段，由于获得了相同的固定点，因此可以肯定的是，后期de Sitter时代不会受到影响。但是，在物质占主导和辐射占主导的阶段，达到的固定点受热项的存在影响，因此，热效应必定会破坏物质和辐射占主导地位的时代。但是，对于大裂口情况，在动力学系统中发现了一些不稳定性，因为初始条件极大地影响了动力学系统在接近起始点时的行为。$Ë$折数的演变。最后，我们通过检查几个状态查找器量来考虑热项对宇宙后期演化的影响。我们的发现表明，只有在对热项的乘法系数进行极微调的情况下，才可以报告这些结果。