Magnetically confined mountains on accreting neutron stars are promising sources of continuous-wave gravitational radiation and are currently the targets of directed searches with long-baseline detectors like the Laser Interferometer Gravitational Wave Observatory (LIGO). In this paper, previous ideal-magnetohydrodynamic models of isothermal mountains are generalized to a range of physically motivated, adiabatic equations of state. It is found that the mass ellipticity ε drops substantially, from ε≈ 3 × 10−4 (isothermal) to ε≈ 9 × 10−7 (non-relativistic degenerate neutrons), 6 × 10−8 (relativistic degenerate electrons) and 1 × 10−8 (non-relativistic degenerate electrons) (assuming a magnetic field of 1012.5 G at birth). The characteristic mass Mc at which the magnetic dipole moment halves from its initial value is also modified, from Mc/M⊙≈ 5 × 10−4 (isothermal) to Mc/M⊙≈ 2 × 10−6, 1 × 10−7, and 3 × 10−8 for the above three equations of state, respectively. Similar results are obtained for a realistic, piecewise-polytropic nuclear equation of state. The adiabatic models are consistent with current LIGO upper limits, unlike the isothermal models. Updated estimates of gravitational-wave detectability are made. Monte Carlo simulations of the spin distribution of accreting millisecond pulsars including gravitational-wave stalling agree better with observations for certain adiabatic equations of state, implying that X-ray spin measurements can probe the equation of state when coupled with magnetic mountain models.