Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X_{t_{0}}=p^{-1}(t_{0})$, for a $t_{0}\in T$, has only canonical singularities of index $1$. Is there a neighborhood $U$ of $X_{t_{0}}$ such that any variety $X_{t}$ belonging to $U$ has at worst canonical singularities? Any variety is over $\mathbb{C}$ here.

When the base of the deformation is smooth the answer is *yes*. This was proven by Kawamata in his paper *Deformations of canonical singularities*, Journal of the American Mathematical Society **12** (1999), 85-92.

For the reader's convenience, let me write here the precise statement of Kawamata's result.

Theorem (Kawamata).Let $\pi \colon X \to S$ be a flat morphism from a germ of a variety $(X, \, x_0)$ to a germ of a smooth curve $(S, \, s_0)$, whose special fiber $X_0 = \pi^{-1}(s_0)$ has only canonical singularities. Then $X$ has only canonical singularities. In particular, the fibers $X_s = \pi^{-1}(s)$ have only canonical singularities.

Note that your assumption about the index of the singularities of $X_{0}$ is unnecessary.