  Neutron stars (NSs) have traditionally been viewed as cold, zero-temperature
entities. However, recent progress in computational methods and theoretical
modelling has opened up the exploration of finite temperature effects, marking
a novel research frontier. This study examines Proto-Neutron Stars (PNSs) using
the BigApple parameter set to investigate their macroscopic properties. Two
approaches are employed: one with constant temperatures (10-50 MeV) and the
other fixing entropy per baryon (S) at predefined levels (S = 1 and S = 2).
Notably, S remains constant with increasing baryon density due to
electron-positron pair formation at finite temperatures. Analysis of PNS
mass-radius profiles, considering neutrino trapping and temperature effects,
reveals flattened curves and expanded radii with increasing temperature,
resulting in slightly higher masses compared to zero temperature. The influence
of lepton fraction ($Y_l$) on maximum PNS mass is explored, indicating that
higher $Y_l$ values lead to a softer Equation of State (EoS), reducing maximum
mass and increasing the canonical radius ($R_{1.4}$). Further investigation of
a constant entropy EoS demonstrates that higher entropy is associated with
increased maximum PNS masses and flatter mass-radius curves. Central
temperature versus maximum mass relationships suggest a correlation between NS
mass and temperature. Lastly, we investigate the behaviour of $f$-mode
frequencies in PNS. It reveals that the frequency of these modes decreases with
increasing entropy and temperature, reflecting complex thermodynamic
interactions within the stars.
