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We study the Scale-covariant theory of gravity \cite{Canuto:1977zz} under thecosmic surrounding of an isotropic flat FLRW space-time. We evaluate thedeceleration parameter, which is a hyperbolic function of the cosmic time $ t$. The deceleration parameter transits from deceleration to acceleration inlate times. In turn, we create the model having the behavior of the late timeuniverse, which is ever accelerated expanding and faces a Big Freeze at theend. The model is transient from a matter-dominated era to quintessence darkenergy in late times. We compute the constrained values of Hubble parameter $H_0=82.537^{+0.903}_{-0.917} $ and the model parameter $n=1.436^{+0.019}_{-0.015} $ using joint analysis of the $ H(z) $ data of77-points and Ia Supernovae data. The model exhibits point-type singularity asit begins with a point of zero volume, infinite energy density, andtemperature. Furthermore, we obtain the present value of deceleration parameter$ (q_0) \approx {-0.536} $. Also, we examine the ultimate behavior of our modelby the proper analysis of energy conditions, cosmographical parameters, andStatefnder diagnostic. Finally, the proposed model behaves like a quintessencedark energy model in late times.

Dynamics of a hyperbolic solution in Scale-covariant theory