Let $f:X\to T$ be a smooth family of complex projective varieties, and assume that the special fiber $X_{t_0}$ is hyperbolic for $t_0\in T$. Is $X_t$ also hyperbolic for all $t\in U$ in a Zariski open neighborhood $U\subseteq T$ of $t_0$?
Here we say that $X$ is (Brody) hyperbolic if there is no non-constant holomorphic map $f:\mathbb C\to X$.
Yes,Hyperbolicity is an open condition .You can find it in Brody's paper in Trans Amer Math Soc vol235 1978 page 216 . A more general statement can be found in Kobayashi's book Hyperbolic Complex Spaces page 148 Theorem 3.11.1
Kobayashi hyperbolicity (or Brody hyperbolicity, the two notions coincide for compact complex spaces), is an open condition with respect to the analytic topology. Thus, for instance, once you find an example of hyperbolic compact complex space, you also get that small deformations are hyperbolic, too. This fact is a direct consequence of the celebrated Brody's lemma. A proof can be found, as already said, in the comprehensive book by Kobayashi or in the Santa Cruz lecture notes by Demailly.
There is another notion of hyperbolicity, which as been introduced by Demailly in his Santa Cruz lecture notes, whose name is algebraic hyperbolicity. This notion is weaker than Kobayashi's one. Roughly speaking, a compact complex space $X$ is algebraically hyperbolic if you can control from below minus the Euler characteristic of (the normalization of) every compact complex curve in $X$ by a (universal constant times) its degree (with respect to some polarization, or more generally, to some hermitian metric).
This notion also satisfies an openness condition. Namely, let $\mathcal X\to S$ be an algebraic family of projective algebraic manifolds (i.e. given by a projective morphism). Then the set of $s\in S$ such that the fiber $X_s$ is algebraically hyperbolic is open with respect to the “countable Zariski topology” of $S$ (by definition, this is the topology for which closed sets are countable unions of algebraic sets).
Quoting Demailly himself, it would be interesting to know whether algebraic hyperbolicity is open with respect to the euclidean topology; still more interesting would be to know whether Kobayashi hyperbolicity is open for the countable Zariski topology (of course, both properties would follow immediately if one knew that algebraic and Kobayashi hyperbolicity coincide, but this seem otherwise highly non trivial to establish).
The latter openness property has raised an important amount of work around the following more particular question: is a (very) generic hypersurface $X\subset\mathbb P^{n+1}$ of degree $d$ large enough (say $d\ge 2n + 1$) Kobayashi hyperbolic? Again, “very generic” is to be taken here in the sense of the countable Zariski topology.
Very recently, this question seems to have been answered in the affirmative in this paper by D. Brotbek. Perhaps, it worths noticing that Brotbek obtains the hyperbolicity for the generic hypersurface, and not only very generic. This might be a peculiar property of families of projective hypersurfaces, but it is not completely clear to me how to fit it in the general picture.
