It could easily happen that $X_0$ has log terminal singularities and $X$ is not log terminal. The standard example is if $f:Y\to X$ is a flipping contraction of a 3-fold over a curve $T$ (where the flipping curve $C$ is contained in the central fiber). The issue is that if $Y_0$ is log terminal, then as $Y_0\to X_0$ is $K_{Y_0}$ negative birational map of surfaces, then $X_0$ also has log terminal singularities (and is even Q-factorial). On the other hand $K_X$ is automatically not Q-Cartier (if it where, then $K_Y=f^*K_X$ and so $K_Y\cdot C=0$ which is impossible). Note however that if you require $X_0$ to be canonical, then so is $X$ (see Theorem 1.4 of arXiv:math/9809091). Finally, if you assume that $X$ is log terminal, you can deduce almost nothing for $X_0$ (it could be reducible or even non-reduced). However, if you assume that $(X,X_0)$ is plt, then you can deduce that $(K_X+X_0)|_{X_0}=K_{X_0}+B_0$ where $(X_0,B_0)$ is klt (the converse to this statement is also true; it is known as inversion of adjunction; see the book of Kollár-Mori).