  A finite-temperature ($T>0$) study of a model of a mixture of spin-zero
hardcore bosons and spinless fermions, with filling fractions $\rho_B$ and
$\rho_F$, respectively, on a two-dimensional square lattice with composite
hopping $t$ is presented. The composite hopping swaps the locations of a
fermion and a boson that occupy nearest-neighbor sites of the lattice. The
superfluid order parameter $\psi$, the femion hopping amplitude $\phi$, the
chemical potential $\mu$, the free energy minimum $\tilde{F}$ and entropy $S$
are calculated in the limit $\rho_B+\rho_F=1$ within a mean-field
approximation, and lead to a phase diagram in the $\rho_F - T$ plane. This
phase diagram consists of a metallic superfluid phase under a dome-shaped
$T(\rho_F)$, and insulating normal liquid and insulating normal gas phases
outside the dome. These phases are separated by coupled discontinuous
transitions as indicated by jumps in $\psi$ and $\phi$. The maximum critical
transition temperature $T_c$ is observed very close to $\rho_F = 1/2$. While
$\tilde{F} (T)$ is continuous with a derivative discontinuity at $T=T_c
(\rho_F)$ for $0 <\rho_F \le 1/2$ (first-order transition), it becomes {\em
discontinuous} for $\rho_F>1/2$ (zeroth-order transition), where the entropy
becomes negative for a range of temperatures below $T_c$. The ratio of $T_c$ to
Fermi band width agrees remarkably with the ratio of $T_c$/$T_F$ (where $T_F$
is the Fermi temperature) of unconventional superfluids and superconductors
like Fermi-Bose mixtures, the high-$T_c$ cuprates, iron-based and hydride
superconductors, that exhibit experimental values of $T_c$ spread over nine
orders of magnitude from $\sim 200$nK to $\sim 260$K.
