In thermodynamics a macroscopic state of a system results from a number of its microscopic states. This number is given by the exponent of the system's entropy exp(S). In noninteracting systems with discrete energy spectra, such as large scale quantum dots, S as a function of the temperature has usually a plateau shape with integer values of exp(S) on these plateaus. Plateaus with noninteger values of exp(S) are fundamentally forbidden and would be thermodynamically infeasible. Here we investigate the entropy of a noninteracting quantum dot coupled via tunneling to normal metals with continuum spectra as well as to topological superconductors. We show that the entropy may have noninteger plateaus if the topological superconductors support weakly overlapping Majorana bound states. This brings a fundamental change in the thermodynamics of the quantum dot whose specific heat c(V) acquires low-temperature Majorana peaks which should be absent according to the conventional thermodynamics. We also provide a fundamental thermodynamic understanding of the transport properties, such as the linear conductance. In general our results show that the thermodynamics of systems coupled to Majorana modes represents a fundamental physical interest with diverse applications depending on versatility of possible coupling mechanisms.