We study the thermodynamical aspects of [Formula: see text] gravity in the Jordan and the Einstein frame, and we investigate the corresponding equivalence of the thermodynamical quantities in the two frames. We examine static spherically symmetric black hole solutions with constant Ricci scalar curvature [Formula: see text], and as we demonstrate, the thermodynamical quantities in the two frames are equivalent. However, for the case of black holes with nonconstant scalar curvature [Formula: see text], the thermodynamical equivalence of the two frames is no longer valid. In addition, we extend our study to investigate cosmological solutions with a homogeneous and isotropic background. In particular, we find that the power-law cosmology case provides an accidentally thermodynamical equivalence of the two frames. However, for nontrivial cosmology, we examine a novel exponential ultraviolet [Formula: see text] gravity. This confirms that the thermodynamical quantities in both frames are not equivalent. In conclusion, although [Formula: see text] gravity and its corresponding scalar-tensor theory are mathematically equivalent, at least for conformal invariant quantities, the two frames are not thermodynamically equivalent at a quantitative level, in terms of several physical quantities.

中文翻译：

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Jordan 和 Einstein 框架中 f(R) 引力的热力学对应

我们在 Jordan 和 Einstein 框架中研究 [公式：见文本] 引力的热力学方面，并研究了两个框架中热力学量的相应等价性。我们检查具有恒定 Ricci 标量曲率的静态球对称黑洞解 [公式：见正文]，正如我们所证明的，两个框架中的热力学量是等价的。然而，对于具有非恒定标量曲率的黑洞[公式：见正文]，两个框架的热力学等价不再有效。此外，我们将研究扩展到研究具有均匀和各向同性背景的宇宙学解决方案。特别是，我们发现幂律宇宙学案例提供了两个框架的意外热力学等价。然而，对于非平凡的宇宙学，我们研究了一种新的指数紫外线[公式：见正文]重力。这证实了两个框架中的热力学量不相等。总之，尽管[公式：见正文]重力及其相应的标量张量理论在数学上是等价的，至少对于共形不变量而言，这两个框架在几个物理量方面在定量水平上并不是热力学等价的。