We investigate the possibility of performing cosmological studies in the redshift range $2.5<z<5$ through suitable extensions of existing and upcoming radio-telescopes like CHIME, HIRAX and FAST. We use the Fisher matrix technique to forecast the bounds that those instruments can place on the growth rate, the BAO distance scale parameters, the sum of the neutrino masses and the number of relativistic degrees of freedom at decoupling, $N_{rm eff}$. We point out that quantities that depend on the amplitude of the 21cm power spectrum, like $fsigma_8$, are completely degenerate with $Omega_{rm HI}$ and $b_{rm HI}$, and propose several strategies to independently constraint them through cross-correlations with other probes. Assuming $5%$ priors on $Omega_{rm HI}$ and $b_{rm HI}$, $k_{rm max}=0.2~h{rm Mpc}^{-1}$ and the primary beam wedge, we find that a HIRAX extension can constrain, within bins of $Delta z=0.1$: 1) the value of $fsigma_8$ at $simeq4%$, 2) the value of $D_A$ and $H$ at $simeq1%$. In combination with data from Euclid-like galaxy surveys and CMB S4, the sum of the neutrino masses can be constrained with an error equal to $23$ meV ($1sigma$), while $N_{rm eff}$ can be constrained within 0.02 ($1sigma$). We derive similar constraints for the extensions of the other instruments. We study in detail the dependence of our results on the instrument, amplitude of the HI bias, the foreground wedge coverage, the nonlinear scale used in the analysis, uncertainties in the theoretical modeling and the priors on $b_{rm HI}$ and $Omega_{rm HI}$. We conclude that 21cm intensity mapping surveys operating in this redshift range can provide extremely competitive constraints on key cosmological parameters.
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