  We study temperature fluctuations in mesoscopic $N$-body systems undergoing
non-equilibrium processes from the perspective of stochastic thermodynamics. By
introducing a stochastic differential equation, we describe the evolution of
the system's temperature during an isothermal process, with the noise term
accounting for finite-size effects arising from random energy transfer between
the system and the reservoir. Our analysis reveals that these fluctuations make
the extensive quantities (in the thermodynamic limit) deviate from being
extensive for consistency with the theory of equilibrium fluctuation. Moreover,
we derive finite-size corrections to the Jarzynski equality, providing insights
into how heat capacity influences such corrections. Also, our results indicate
a possible violation of the principle of maximum work by an amount proportional
to $N^{-1}$. Additionally, we examine the impact of temperature fluctuations in
a finite-size quasi-static Carnot engine. We show that irreversible entropy
production resulting from the temperature fluctuations of the working substance
diminishes the average efficiency of the cycle as $\eta_{\rm{C}}-\left\langle
\eta\right\rangle \sim N^{-1}$, highlighting the unattainability of the Carnot
efficiency $\eta_{\rm{C}}$ for mesoscopic-scale heat engines even under the
quasi-static limit
