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In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions. • A fully fractional thermostat model involving the Riemann–Liouville derivatives. • The equivalence between the FDE and the corresponding integral equation is obtained. • Some new existence and uniqueness results are established. • An implicit numerical scheme based on the right product rectangle rule is derived. • Numerical simulations of implicit solutions are presented. [ABSTRACT FROM AUTHOR]