This paper is based on analysis of cosmological parameters and thermodynamics in a modified theory of gravity whose action contains linear coupling of Ricci scalar $R$ and a second degree term of its covariant derivative. The Friedmann equations obtained in result of considering the spatially flat FRW universe filled with dark energy and pressureless matter, are of fifth-order in the Hubble function. Three different deceleration parameters $q$ are used in an integral relation to obtain the Hubble parametric models $H\left(z\right)$. We explore the equation of state parameter ${\omega }_{DE}$, the behavior of phase plane ${\omega }_{DE}-{\omega }_{DE}^{\prime }$ and the stability of each Hubble parametric model $H\left(z\right)$ via squared speed of sound. The equation of state parameter indicates phantom era of the universe, the phase plane ${\omega }_{DE}-{\omega }_{DE}^{\prime }$ lies in the range of freezing region of the accelerated universe while $v_s^2$ gives the stability of these models under choice of some specific constant parameters. For these models, we also examine the viability of generalized second law of thermodynamics on the apparent horizon with Bekenstein entropy and find that for two models, this law is verified.

本文基于修正的引力理论中的宇宙学参数和热力学分析，引力理论包含Ricci标量的线性耦合 $\mathrm{[R}$以及其协变导数的二阶项 考虑到充满了暗能量和无压物质的空间平坦的FRW宇宙而获得的Friedmann方程在哈勃函数中处于五阶。三种不同的减速参数$q$ 用于积分关系以获得哈勃参数模型 $H（ž）$。我们探索状态参数方程${\omega }_{dË}$，相平面的行为 ${\omega }_{dË}-{\omega }_{dË}^{\prime }$ 和每个哈勃参数模型的稳定性 $H（ž）$通过声音的平方速度。状态方程式表示宇宙的幻影时代，相平面${\omega }_{dË}-{\omega }_{dË}^{\prime }$ 处于加速宇宙的冰冻区域范围内，而 $v_s^2$在选择某些特定常数参数的情况下给出了这些模型的稳定性。对于这些模型，我们还使用Bekenstein熵在视域上研究了热力学广义第二定律的可行性，并发现对于两个模型，该定律得到了验证。