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We study the large eigenvalue asymptotics for the Schro¨dinger operator HV=−1/2Δ+V on the real and the complex hyperbolic n-spaces. Here Δ is the Laplace–Beltrami operator and V is a scalar potential. We assume that V is real-valued, continuous, semi-bounded from below and diverges at infinity in an appropriate sense. Then it is proven that the number of eigenvalues of HV less than λ behaves semi-classically as λ↗∞. This is a natural generalization of the result obtained by Inahama and Shirai [Eigenvalue asymptotics for the Schro¨dinger operators on the hyperbolic plane, J. Funct. Anal., submitted for publication]. [Copyright &y& Elsevier]