We study the thermodynamics and thermodynamic geometry of a five-dimensional Reissner-Nordström-AdS black hole in the extended phase space by treating the cosmological constant as being related to the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. We find that the contribution of the charge of the black hole to the chemical potential is always positive, and the existence of charge makes the chemical potential become positive more easily. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric, and Quevedo metric, respectively, in the fixed $N^2$ case and the fixed $q$ case. We find that in the fixed $N^2$ case, the divergence of the scalar curvature is related to the divergence of the specific heat with fixed electric potential in the Weinhold metric and Ruppeiner metric, and the divergence of the scalar curvature in the Quevedo metric corresponds to the divergence of the specific heat with fixed electric charge density. In the fixed $q$ case, however, the divergence of the scalar curvature is related to the divergence of the specific heat with fixed chemical potential in the Weinhold metric and Ruppeiner metric, while in the Quevedo metric, the divergence of the scalar curvature corresponds to the divergence of the specific heat with a fixed number of colors and the vanishing of the specific heat with a fixed chemical potential.

We study the thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in AdS ₅ × S ⁵ spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we find that the scalar curvature in the Weinhold metric is always vanishing, while in the Ruppeiner metric the divergence of the scalar curvature is related to the divergence of the heat capacity with fixed chemical potential, and in the Quevedo metric the divergence of the scalar curvature is related to the divergence of the heat capacity with fixed number of colors and to the vanishing of the heat capacity with fixed chemical potential.

We propose a new single field inflation model in which the usual slow-roll inflation is joined to a new period of slow-climb and slow-roll inflation through a short intermediate noninflationary phase. We then show that primordial curvature perturbations can be enhanced at small scales, a sizable amount of primordial black holes (PBHs) can be produced which make up most of dark matter, the gravitational waves (GWs) induced by scalar metric perturbations that accompany with the formation of PBHs can be detectable by future GW experiments, and last but not least, our model is compatible with the latest cosmic microwave background observations. Remarkably, the GW spectrum displays a unique oscillating character in the ultraviolet regions which originates from the short noninflationary phase. A detection of such oscillations in the GW spectrum may suggest the existence of such a noninflationary phase in the whole inflation, thus providing us a chance to reveal an interesting period in the evolution of the early Universe and distinguish our model from others.

We study thermodynamics of the four-dimensional Kerr-Sen-AdS black hole and its ultraspinning counterpart and verify that both black holes fulfil the first law and Bekenstein-Smarr mass formulas of black hole thermodynamics. Furthermore, we derive new Christodoulou-Ruffini-like squared-mass formulas for the usual and ultraspinning Kerr-Sen-${\mathrm{AdS}}_4$ solutions. We show that this ultraspinning Kerr-Sen-${\mathrm{AdS}}_4$ black hole does not always violate the reverse isoperimetric inequality (RII) since the value of the isoperimetric ratio can be larger/smaller than, or equal to unity, depending upon where the solution parameters lie in the parameters space. This property is obviously different from that of the Kerr-Newman-${\mathrm{AdS}}_4$ superentropic black hole, which always strictly violates the RII, although both of them have some similar properties in other aspects, such as horizon geometry and conformal boundary. In addition, it is found that while there exists the same lower bound on mass ($m_e\ge 8l/\sqrt{27}$ with $l$ being the cosmological scale) both for the extremal ultraspinning Kerr-Sen-${\mathrm{AdS}}_4$ black hole and for the extremal superentropic Kerr-Newman-${\mathrm{AdS}}_4$ case, the former has a maximal horizon radius $r_{\mathrm{HP}}=l/\sqrt{3}$, which is the minimum of the latter. Therefore, these two different kinds of four-dimensional ultraspinning charged AdS black holes exhibit some significant physical differences.

We explore some (especially, thermodynamical) properties of the dyonic Kerr-Sen-${\mathrm{AdS}}_4$ black hole and its ultraspinning counterpart, and check whether or not both black holes satisfy the first law and Bekenstein-Smarr mass formulas. To this end, new Christodoulou-Ruffini-like squared-mass formulas for the usual dyonic Kerr-Sen-${\mathrm{AdS}}_4$ solution and its ultraspinning cousin are deduced. Similar to the ultraspinning Kerr-Sen-${\mathrm{AdS}}_4$ black hole case, we demonstrate that the ultraspinning dyonic Kerr-Sen-${\mathrm{AdS}}_4$ black hole does not always violate the reverse isoperimetric inequality (RII) since the value of the isoperimetric ratio can either be larger/smaller than, or equal to unity, depending upon the range of the solution parameters, as is the case only with an electric charge. This property is apparently distinct from that of the superentropic dyonic Kerr-Newman-${\mathrm{AdS}}_4$ black hole, which always strictly violates the RII, although both of them have some similar properties in other aspects, such as the horizon geometry and conformal boundary.