In this work, we have analysed the Barrow holographic dark energy model in the Brans–Dicke cosmology in the framework of the flat Friedmann–Lemaitre–Robertson–Walker universe. To discuss the cosmological implications of the model, we have considered a well motivated logarithmic form of Brans–Dicke scalar field. Barrow holographic dark energy is based on Barrow entropy rather than the standard Bekenstein–Hawking entropy. The component \(\Delta \) with bounds \(0\le \Delta \le 1\) is the quantum-gravitational distortion which modifies holographic dark energy. The cosmological parameters like the equation of state parameter and the deceleration parameter are obtained to explain the evolution of the universe by using the various estimations of Barrow exponent \(\Delta \) and other model parameters. We have observed that the equation of state parameter shows quintessence like behaviour at present and in future. The deceleration parameter shows recent phase transition of the universe (decelerated to accelerated expansion). Further, deceleration parameter shows decelerated expansion of the universe in future. We have discussed cosmic coincidence problem and found that it does not appear in the present model. The \(w_h-w_h^\prime \) analysis of the model shows its similarity with some well known DE models in different time periods, however, over all evolution of the trajectories is quite different in comparison of well known DE models. Further, we show that our model satisfies the generalized second law of thermodynamics for suitable choice of model parameters.

在这项工作中，我们在平坦的弗里德曼-勒梅特-罗伯逊-沃克宇宙的框架内分析了布兰斯-迪克宇宙学中的巴罗全息暗能量模型。为了讨论该模型的宇宙学含义，我们考虑了布兰斯-迪克标量场的良好动机对数形式。巴罗全息暗能量基于巴罗熵而不是标准的贝肯斯坦-霍金熵。边界为\ (0\le \Delta \le 1\ ) 的分量 \(\Delta \)是修改全息暗能量的量子引力畸变。利用巴罗指数\(\Delta \)和其他模型参数的各种估计，得到状态方程参数、减速参数等宇宙学参数来解释宇宙的演化。我们观察到，状态参数方程显示了当前和未来的典型行为。减速参数显示了宇宙最近的相变（减速到加速膨胀）。此外，减速参数表明未来宇宙的膨胀速度会减速。我们讨论了宇宙巧合问题，发现它没有出现在当前的模型中。该模型的\(w_h-w_h^\prime \)分析显示其与不同时期的一些著名的DE模型相似，但总体轨迹的演化与众所周知的DE模型相比有很大不同。此外，我们表明我们的模型满足广义热力学第二定律，可以适当选择模型参数。