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.
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