In recent years there has been growing interest in structure-preserving numerical methods, namely methods designed to reproduce physical properties and symmetries of the governing equations at the discrete level. While remarkable advances have been achieved for spatial discretizations [1], often neglecting temporal discretization errors, time-integration schemes can also be tailored to enforce structural properties. Moreover, whereas structure-preserving spatial discretizations are generally designed to operate independently of grid resolution, the idea of an adaptive time-stepping strategy that provides a user-defined trade-off between physical fidelity and computational cost also requires further investigation and may prove particularly advantageous in the context of high-fidelity, large-scale simulations. For incompressible flows, an efficient adaptation strategy has been developed to guarantee time-induced kinetic-energy conservation up to a specified tolerance [2]. In this work, we aim at extending such principles to generic conservative or strictly dissipative systems which determines the evolution of a convex variable referred to as the entropy such for systems. The entropy function can either remain constant in time (for conservative systems) or decrease monotonically (for dissipative systems). We propose a general mathematical framework that is applicable to any physical system that can be written in conservative or fully dissipative form. Thus, its effectiveness will be demonstrated through multiphysics test cases, by also combining it with the most recent advances in the framework of conservative spatial discretizations [3]. [1] A.E.P. Veldman, Supra-conservative finite-volume methods for the simulation of subsonic flow, 14th WCCM ECCOMAS Congress, Paris, France, 2020. [2] F. Capuano, B. Sanderse, E. M. De Angelis, G. Coppola, A minimum-dissipation time-integration strategy for large-eddy simulation of incompressible turbulent flow, XXIII Aimeta Congress, Salerno, Italy, 2017. [3] A. Aiello, C. De Michele, G. Coppola, Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state, Journal of Computational Physics, Vol. 528, 113836, 2025.