I have a log pair $(X,D)$ which is purely log terminal and $D$ is a projective divisor ($X$ may not be projective). Moreover, $D$ is a variety of Fano type. Is there a space of finite type parametrizing first order deformation theory of the pair $(X,D)$ which are isotrivial along $D$? i.e. the induced flat deformation on $D$ is isotrivial.