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We study the heat statistics of a multi-level $N$-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number $M$ of measurements $(M \to \infty)$. In this context, the conditions allowing for an Infinite-Temperature Thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the non-equilibrium evolution of the system and its initial state. Exceptions to ITT, to which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasi-commuting) with the Hamiltonian of the quantum system, or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ($N \to \infty$), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits $M\to\infty$ and $N\to\infty$ matters: when $N$ is fixed and $M$ diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A non trivial result is obtained fixing $M/N^2$ where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.