Deformations of the tangent bundle of a projective hypersurface

For a nonsingular hypersurface $X\subset {\mathbb{P}}^n,n\ge 4,$ of degree $d\ge 2$, we show that the space $H^1(X,\text{End}(T_X))$ of infinitesimal deformations of the tangent bundle $T_X$ has dimension $\left(\genfrac{}{}{0px}{}{n+d-1}{d}\right)(d-1)$ and all infinitesimal deformations are unobstructed even though $H^2(X,\text{End}(T_X))$ can be nonzero. Furthermore, we prove that the irreducible component of the moduli space of stable bundles containing the tangent bundle is a rational variety, by constructing an explicit birational model.