We study the behaviors of mean end-to-end distance and specific heat of a two-dimensional intrinsically curved semiflexible biopolymer with a hard-core excluded volume interaction. We find that the mean square end-to-end distance RN2∝Nβ at large N, with N being the number of monomers. Both β and proportional constant are dependent on the reduced bending rigidity κ and intrinsic curvature c. The larger the c, the smaller the proportional constant, and 1.5≥β≥1. Up to a moderate κ=κc, or down to a moderate temperature T=Tc, β=1.5, the same as that of a self-avoiding random walk, and the larger the intrinsic curvature, the smaller the κc. However, at a large κ or a low temperature,βis close to 1, and the conformation of the biopolymer can be more compact than that of a random walk. There is an intermediate regime with 1.5 > β > 1 and the transition from β=1.5 to β=1 is smooth. The specific heat of the system increases smoothly with increasing κ or there is no peak in the specific heat. Therefore, a nonvanishing intrinsic curvature seriously affects the thermal properties of a semiflexible biopolymer, but there is no phase transition in the system.