In this article, thermal buckling and natural frequency of a curved functionally graded (FG) nanobeam in a thermal environment based on Eringen’s theory is investigated. Dimension of structure is in small scale, its geometric is curved, and properties of material vary in radial direction. In order to develop differential equation and boundary condition, Hamilton’s principle is adopted. Properties of material are a function of two variables of radial thickness and temperature. After developing equation of motion in thermal environment, analytical solution has been employed in order to obtain the amount of frequency and thermal buckling. Free vibration of a curved FG nanobeam subjected to in-plane thermal load may show zero frequency magnitude at a certain temperature, which specifies the existence of bifurcation type of instability. In numerical section, frequency responses have been studied one time based on temperature-dependent material property and another time based on temperature-independent material property and influences for parameters such as nonlocal parameter, power-law, mode number, temperature changes, and arc angle on natural frequency and critical temperature have been investigated. Results have shown that if properties of material are dependent on temperature, then expected frequency will be less than the case in which properties are independent of temperature. Performed validation certifies correctness of obtained results. Results indicate that critical temperature increasing the arc angle leads to a decrease in amount of dimensionless frequency, and this matter represents the importance of specification of critical temperature in curved structures.