Line bundle on a product

Let $p:X \times T \to X$ and $q:X \times T \to T$ the canonical projections.

Hint 1: show that for every $t \in T$ there is an open set $t \in U_t \subseteq T$, such that all $\mathcal{L}_{t'}$ with $t' \in U_t$ are isomorphic to $\mathcal{L}_{t}$.

Hint 2: to prove this, consider $\mathcal{L}_0 = \mathcal{L}|_{X \times t}$ as a line bundle on $X$ and consider $\mathcal{M} = (p^* \mathcal{L}_0)^{-1} \otimes \mathcal{L}$. The line bundle $\mathcal{M}$ has $\mathcal{O}_X$ as the fiber over $t$.

Hint 3: By using the semicontinuity theory of Hartshorne prove, that $q_*\mathcal{M}$ is locally free on a certain open neighbourhood of $t \in T$. (here $H^1(X,\mathcal{O}_X) = 0$ is used).

Hint 4: use the result of Hint 3 to prove that for all $t'$ in a certain open neighbourhood $V$ of $t$ the fibers $\mathcal{M}|_{X \times t'}$ are isomorphic to $\mathcal{O}_X$.

If you want to see a solution you can look at Kommutative Algebra und algebraische Geometrie, p.392 (book is in German)

Let $X$ be an integral projective scheme over an algebraically closed field $k$, and assume that $H^1(X, \mathcal{O}_X) = 0$. Let $T$ be a connected scheme of finite type over $k$. We want to show that if $\mathscr{L}$ is an invertible sheaf on $X \times T$, then the invertible sheaves $\mathscr{L}_t$ on $X = X \times \{t\}$ are isomorphic, for all closed points $t \in T$. This is the content of Exercise III.12.6(a) of Hartshorne, as mentioned in the comments above.

We have a (group) scheme $\text{Pic}(X)$ which essentially parametrizes line bundles on $X$ (and which happens to be an infinite disjoint union of projective schemes, although that's not terribly relevant). $H^1(X, \mathcal{O}_X)$ is the tangent space of $\text{Pic}(X)$ (at every point), so if it is $0$, it follows that $\text{Pic}(X)$ is actually a disjoint union of (reduced) points. Then $\mathscr{L}$ induces a morphism $T$ to $\text{Pic}(X)$, which must be constant since $T$ is connected and $\text{Pic}(X)$ is a disjoint union of points. Because the morphism $T$ to $\text{Pic}(X)$ is constant, the fibers $\mathscr{L}_t$ are all isomorphism (we have to be slightly careful — morphisms $T$ to $\text{Pic}(X)$ are not in bijection with line bundles of $X \times T$, but any two line bundles inducing the same morphism will have isomorphic fibers $\mathscr{L}_t$). For details on this material, look at the book "Néron models" here.

Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel. Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21. Springer-Verlag, Berlin, 1990. x+325 pp.