  Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations
$\pi$ of $\mathrm{GL}_n$ with unitary central character over a number field
$F$. We prove the first unconditional zero density estimate for the set
$\mathcal{S}=\{L(s,\pi\times\pi')\colon\pi\in\mathfrak{F}_n\}$ of
Rankin-Selberg $L$-functions, where $\pi'\in\mathfrak{F}_{n'}$ is fixed. We use
this density estimate to prove (i) a strong average form of effective
multiplicity one for $\mathrm{GL}_n$; (ii) that given $\pi\in\mathfrak{F}_n$
defined over $\mathbb{Q}$, the convolution $\pi\times\tilde{\pi}$ has a
positive level of distribution in the sense of Bombieri-Vinogradov; (iii) that
almost all $L(s,\pi\times\pi')\in \mathcal{S}$ have a hybrid-aspect
subconvexity bound on $\mathrm{Re}(s)=\frac{1}{2}$; (iv) a hybrid-aspect
power-saving upper bound for the variance in the discrepancy of the measures
$|\varphi(x+iy)|^2 y^{-2}dxdy$ associated to $\mathrm{GL}_2$ Hecke-Maass
newforms $\varphi$ with trivial nebentypus, extending work of Luo and Sarnak
for level 1 cusp forms; and (v) a nonsplit analogue of quantum ergodicity:
almost all restrictions of Hilbert Hecke-Maass newforms to the modular surface
dissipate as their Laplace eigenvalues grow.
