This paper is devoted to the obtaining of the exact solutions in the framework of modified $f(R)$-gravity's rainbow(UV completion of General Relativity)/$f(R)$-gravity and the study of their thermodynamic behavior. In the $f(R)$-gravity's rainbow, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form (we should note that scalar field $\mathrm{\Psi }(\mathbf{x})$ couples to gravity minimally or non-minimally with coupling constant λ and the gravity representation which is coupled with the scalar field is $f(R)$ not just R). The most important result derived from this study is that gravity's rainbow increases the height of this potential in the exterior region of the event horizon. We also figure out the effect of the coupling constant λ and the $f(R)$ parameter η on the height of the potential barrier and the other thermodynamical quantities like temperature and heat capacity. We have shown that the effect of $f(R)$-gravity on the temperature adds a coefficient of 1/2 to the cosmological constant.¹ We study the quasinormal modes (QNMs) of 3D black holes in the $f(R)$-gravity's rainbow. For this purpose, we use the WKB approximation method upto third order corrections. We have shown the perturbations decay in corresponding diagrams when the coupling constant λ and rainbow function $g(\epsilon )$ change. We also obtain two different solutions when Ricci scalar is constant. The first one is for a charged slowly rotating toroidal black holes and the second one is for a higher dimensional toroidal black holes inspired by non-commutative geometry. The source for the second one is given by a fluid-type matter with the Gaussian-distribution smeared mass density. In Starobinsky model we show that the charged slowly rotating solution is ill-defined for s²=1 value.

本文致力于在修正的框架内获得精确解。 $F(\mathrm{电阻})$-重力彩虹（广义相对论的UV补全）/$F(\mathrm{电阻})$-重力及其热力学行为的研究。在里面$F(\mathrm{电阻})$-引力彩虹，我们已经通过将径向运动方程转化为标准薛定谔形式来分析有效势垒的概念（我们应该注意标量场 $\mathrm{\Psi }(\mathbf{X})$以耦合常数λ最小或非最小地耦合到重力，与标量场耦合的重力表示为$F(\mathrm{电阻})$不仅仅是R）。从这项研究中得出的最重要的结果是，引力彩虹增加了事件视界外部区域中这种势能的高度。我们还计算了耦合常数λ和$F(\mathrm{电阻})$势垒高度上的参数η和其他热力学量，如温度和热容。我们已经证明了$F(\mathrm{电阻})$- 温度的重力为宇宙常数增加了 1/2 的系数。¹我们研究了 3D 黑洞的准正规模 (QNM)$F(\mathrm{电阻})$-重力的彩虹。为此，我们使用 WKB 近似方法进行三阶校正。我们已经在相应的图表中展示了当耦合常数λ和彩虹函数$G(\epsilon )$改变。当 Ricci 标量恒定时，我们也获得了两种不同的解。第一个是带电缓慢旋转的环形黑洞，第二个是受非交换几何启发的高维环形黑洞。第二个的来源是一种流体类型的物质，具有高斯分布的涂抹质量密度。在 Starobinsky 模型中，我们表明带电缓慢旋转的解对于 s ² =1 值是不明确的。