## 1 Introduction
On the other hand, cyclic loading significantly affects the physical properties and mechanical response of granite. Recent findings indicated that the fatigue failure of rocks can be categorized into three distinct stages: the initial deformation stage, the constant deformation stage, and the accelerated deformation stage, as detailed in the studies of [PERSON] et al. [23; 24]. Extensive research has been conducted to investigate the impact of cyclic loading on the fracture toughness of rocks [25; 26; 27; 28; 29]. The results indicated that the fracture toughness of the specimens experiences a noticeable decrease when subjected to cyclic loading. Regarding cyclic loading frequency, [PERSON] et al. [30] and [PERSON] et al. [31] examined the connection between cyclic loading frequency and crack propagation. Their findings demonstrated that at higher frequencies, such as 1 Hz, cracks exhibit rapid growth in terms of length, number, and width. In more recent research, [PERSON] et al. [32] conducted a comprehensive analysis of the connection between dynamic fracturing, failure modes, and acoustic emission signals in marble subjected to freeze-thaw and stress cycles. As such, several factors are found to affect the rock fatigue mechanical property, including maximum stress, stress amplitude, and loading waveform, as demonstrated by [PERSON] et al. [33] and [PERSON] et al. [34; 35]. The investigation into the fatigue mechanical behavior of granite at real-time temperatures remains relatively overlooked in the current literature. However, in the environment of cyclic hydraulic fracturing, HDRs experience a coupled environment characterized by high temperatures and cyclic loading. Consequently, this study presents a novel approach by conducting cyclic loading and unloading experiments under real-time temperature con ditions, aiming to simulate actual operational scenarios more effectively. In this regard, the present study focuses on the examination of fracture mechanisms and the alterations in the initial load of granite, with a particular emphasis on changes in mechanical properties such as cumulative vertical displacement (CVD), elastic stiffness, fracture toughness, and the reduction in fracture toughness under the combined effects of cyclic loading and real-time high temperatures. In recent years, there has been a growing demand for extracting HDR resources at depths shallower than 5 km, with temperatures ranging from 100 \({}^{\circ}\)C to 400 \({}^{\circ}\)C. This shift is driven by economic considerations and equipment limitations, as noted by [PERSON] et al. (2016). To achieve this, we conducted three-point bending (CTB) tests on granite samples under various real-time high-temperature conditions and implemented different upper load limits, ranging from 75% to 90% of the peak load. We investigated the alterations in load-displacement curves, CVD, elastic stiffness, and fracture toughness, comparing them with results from monotonic three-point bending (MTB) tests. Subsequently, the damage morphology after CTB tests was observed using scanning electron microscopy (SEM). Lastly, we conducted an in-depth analysis of the influence of high temperature and cyclic loading on the crack initiation pressure in three-point bending tests. The findings of this research provide novel perspectives on the fatigue fracturing processes of granite within deep geothermal systems.