  We derive non-flat cosmological models for two cases (i.e., dust and
radiation) in the context of M{\o}ller's tetradic theory (MTT) of gravitation
using the tetrad that creates the non-flat Friedmann-Robertson-Walker (FRW)
metric. These two models are affected by the free dimensional parameter,
$\lambda$, that characterized MTT, which approaches zero in the flat case for
both models. Using standard definitions of thermodynamics, we calculate the
radius horizon, Hawking temperature, and entropy of our non-flat models in the
framework of cosmology and show the effect of $\lambda$ on open and closed
universes. We then use the first law of thermodynamics to construct non-flat
cosmological models via the non-extensive thermodynamic approach. The resulting
models are affected by $\lambda$ and the extensive parameter, $\delta$, which
quantifies the effect of non-extensive thermodynamics. When we set, $\lambda=0$
and $\delta=1$, we return to Einstein's general relativity models. We study the
evolution of our models in the presence of collisionless non-relativistic
matter and describe precise forms of the dark energy density and
equation-of-state parameter constraining the non-extensive thermodynamic
parameter. We show that insertion of the non-extensive thermodynamic parameter
affects the non-flat FRW universe in a manner that noticeably differs from that
observed under normal thermodynamics. We also show that the deceleration of the
open universe behaves as dark energy in a future epoch, i.e., when the redshift
approaches -1, i.e., $z\approx$-1.
