  The emergence of cosmic space as cosmic time progresses is an exciting idea
advanced by Padmanabhan to explain the accelerated expansion of the universe.
The generalization of Padmanabhan's conjecture to the non-flat universe has
resulted in scepticism about the choice of volume such that the law of
emergence can not be appropriately formulated if one uses proper invariant
volume. The deep connection between the first law of thermodynamics and the law
of emergence \cite{mahith}, motivate us to explore the status of the first law
in a non-flat universe when one uses proper invariant volume. We have shown
that the first law of thermodynamics, $dE = TdS +WdV$ cannot be formulated
properly for a non-flat universe using proper invariant volume. We have also
investigated the status of the first law of the form $-dE = TdS$ in a non-flat
universe. We have shown that the energy change dE within the horizon and the
outward energy flux are not equivalent to each other in a non-flat universe
when we use the proper invariant volume. We have further shown that the
consistency between the above two forms of the first law claimed in Ref.
\cite{caiakb} will hold only with the use of the areal volume of the horizon.
Thus, a consistent formulation of the above two forms of the first law of
thermodynamics demands the use of areal volume.
