Projective bundle formula for sheaf cohomology

Question

Given a projective bundle $p:P(E)\rightarrow X$ associated to a vector bundle $E$ and a line bundle on $P(E)$ of the form $\mathcal{O}_{P(E)}(1)\otimes p^*L$ where $L$ is a line bundle on $X$, is there a description of the cohomology groups $H^i(P(E),\mathcal{O}_{P(E)}(1)\otimes p^*L )$ in terms of the cohomology groups of $L$, similar to the projective bundle formulas?

Answer 1

One has $$ Rp_*(\mathcal{O}_{\mathbb{P}(E)}(1) \otimes p^*L) \cong Rp_*(\mathcal{O}_{\mathbb{P}(E)}(1)) \otimes L \cong E^\vee \otimes L, $$ therefore $$ H^\bullet(\mathbb{P}(E), \mathcal{O}_{\mathbb{P}(E)}(1) \otimes p^*L) \cong H^\bullet(X, E^\vee \otimes L). $$