  Using a Coulomb gas technique, we compute analytically the probability
$\mathcal{P}_\beta^{(C)}(N_+,N)$ that a large $N\times N$ Cauchy random matrix
has $N_+$ positive eigenvalues, where $N_+$ is called the index of the
ensemble. We show that this probability scales for large $N$ as
$\mathcal{P}_\beta^{(C)}(N_+,N)\approx \exp\left[-\beta N^2
\psi_C(N_+/N)\right]$, where $\beta$ is the Dyson index of the ensemble. The
rate function $\psi_C(\kappa)$ is computed in terms of single integrals that
are easily evaluated numerically and amenable to an asymptotic analysis. We
find that the rate function, around its minimum at $\kappa=1/2$, has a
quadratic behavior modulated by a logarithmic singularity. As a consequence,
the variance of the index scales for large $N$ as $\mathrm{Var}(N_+)\sim
\sigma_C\ln N$, where $\sigma_C=2/(\beta\pi^2)$ is twice as large as the
corresponding prefactor in the Gaussian and Wishart cases. The analytical
results are checked by numerical simulations and against an exact finite $N$
formula which, for $\beta=2$, can be derived using orthogonal polynomials.
