| Summary: | The general relativity vacuum black holes (BHs) can be scalarised in models where a scalar field non-minimally couples to the Gauss-Bonnet (GB) invariant. Such GB scalarisation comes in two flavours, depending on the GB sign that triggers the phenomenon. Hereafter these two cases are termed GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>±</mo></msup></semantics></math></inline-formula> scalarisation. For vacuum BHs, only GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> scalarisation is possible in the static case, while GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>−</mo></msup></semantics></math></inline-formula> scalarisation is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>p</mi><mi>i</mi><mi>n</mi></mrow></semantics></math></inline-formula> induced. But for electrovacuum BHs, GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>−</mo></msup></semantics></math></inline-formula> is also <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>h</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula> induced. We discuss the GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>−</mo></msup></semantics></math></inline-formula> scalarisation of Reissner-Nordström and Kerr-Newman BHs, discussing zero modes and constructing fully non-linear solutions. Some comparisons with GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> scalarisation are given. To assess the generality of the observed features, we also briefly consider the GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>±</mo></msup></semantics></math></inline-formula> scalarisation of stringy dilatonic BHs and coloured BHs which provide qualitative differences with respect to the electrovacuum case, namely on the distribution and existence of regions triggering GB<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>−</mo></msup></semantics></math></inline-formula> scalarisation.
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