## 1 Introduction  
Geothermal energy has been identified as a highly efficient and reliable form of renewable energy, often referred to as 'green energy', due to its ability to maintain a consistent output regardless of seasonal or diurnal variation [1,2]. The term 'hot dry rock' (HDR) refers to rock formations found at depths of 3-10 km and characterized by high temperatures, which account for a significant proportion of geothermal resources [3,4]. The permeability of HDR reservoirs is primarily enhanced by hydraulic fracturing, creating enhanced geothermal systems (EGSs) that provide efficient heat extraction pathways [5]. However, fractures created by conventional hydraulic fracturing are typically large and singular, which may not result in the desired increase in production. In contrast, liquid nitrogen fracturing has been shown to improve the effectiveness of reservoir fracturing [6]. [PERSON] et al. [7] observed via SEM scanning that, compared with conventional hydraulic fracturing fluids, nitrogen fracturing significantly increased the total crack length, overall fracture surface area, and flow conductivity of the rock, primarily due to the significant temperature difference between the low-temperature liquid nitrogen and the reservoir matrix. This induces numerous microcracks in the matrix under cold shock, resulting in the formation of a complex network of channels. Consequently, there is an imperative for comprehensive studies to be conducted on the evolution of the physical and mechanical properties of high-temperature granite after exposure to liquid nitrogen cold shock. In recent years, a considerable body of research has been dedicated to investigating the alterations in the physical and mechanical properties of granite subjected to thermal treatment [8, 9, 10, 11, 12, 13]. For instance, [PERSON] et al. [14] investigated the mechanical properties of granite under the combined influence of temperature and loading rate. The experimental findings demonstrated that compressive strength and elastic modulus initially increased and then decreased with temperature, reaching a maximum at 200 \({}^{\circ}\)C. When the temperature exceeded 600 \({}^{\circ}\)C, the effect of loading rate on compressive strength diminished, and the failure mode of the rock transitioned from brittle to semi-brittle. In a comparative study on the effects of two cold-shock methods (natural cooling and water-cooling shock) on the mechanical properties of granite after high-temperature heating, [PERSON] et al. [15] found that peak displacement significantly decreased under water-cooling shock, and the crack damage threshold was reached earlier. The study also found that water-cooling shock exacerbated strength degradation by inducing the expansion of intra-grain cracks. [PERSON] et al. [16] conducted a systematic investigation into the mechanical properties of Indian granite under elevated temperatures (30-160 \({}^{\circ}\)C), encompassing uniaxial compressive strength, tensile strength, Poisson's ratio, and elastic modulus. Their research revealed that, under high-temperature conditions, granite exhibited significant anisotropy in its coefficient of thermal expansion. Furthermore, the use of scanning electron microscopy (SEM) images revealed that mineral particle expansion at 160 \({}^{\circ}\)C led to partial crack closure. [PERSON] et al. [17] conducted physical and mechanical tests on granite following its natural cooling from 25 \({}^{\circ}\)C to 1000 \({}^{\circ}\)C. They identified 400 \({}^{\circ}\)C, 600 \({}^{\circ}\)C, and 800 \({}^{\circ}\)C as characteristic temperatures, which they suggest mark distinct stages in the evolution of the physical and mechanical properties of granite. In a related study, [PERSON] et al. [18] performed triaxial compression tests on granite subjected to high-temperature water-cooling shock. Their findings revealed that the Young's modulus and cohesion of granite initially increased and then decreased with increasing temperature. [PERSON] et al.